Microbundles over topological rings

TL;DR

This paper explores the theory of microbundles over topological rings, focusing on their structure, morphisms, automorphisms, extensions, and their compactifications and inverse spectra.

Contribution

It provides a comprehensive study of microbundles over topological rings, including new insights into their structure and classification.

Findings

Characterization of microbundle homomorphisms and automorphisms

Development of compactification techniques for microbundles

Analysis of inverse spectra of microbundles over topological rings

Abstract

The article is devoted to microbundles over topological rings. Their structure, homomorphisms, automorphisms and extensions are studied. Moreover, compactifications and inverse spectra of microbundles over topological rings are investigated.

Full text

address: Dep. Appl. Mathematics, Moscow State Techn. University MIREA, av. Vernadsky 78, Moscow, 119454, Russia

e-mail: sludkowski@mail.ru

1 Introduction.

Microbundles being generalizations of topological manifolds and geometric bundles compose a large area in topology and algebraic topology [4, 12, 17]. Though microbundles on ${\bf R}^{n}$ and Banach spaces over $\bf R$ were investigated, but on topological modules over rings they were not broadly studied. On the other hand, topological rings and topological fields other than $\bf R$ and $\bf C$ are important not only in algebraic topology and general topology, but also in their applications (see, for example, [1, 2, 7]-[11, 13, 14, 18, 19] and references therein). Investigations of microbundles over topological rings are motivated by problems of general topology, algebraic topology, algebraic geometry, representation theory, bundles over topological groups and group rings [6], mathematical physics.

This article is devoted to investigations of microbundles over topological rings. Relations between microbundle and manifold structures are elucidated in Theorems 19, 20 and Corollaries 21, 23-25. In Lemmas 14, 15 and Proposition 16 extensions of microbundles are studied. Inverse spectra of microbundles are investigated in Theorems 27, 29, 30, Propositions 31 and 38. Extensions and homomorphisms of microbundles related with extensions and homomorphisms of topological modules and topological rings are described in Theorem 32 and Corollary 33. Compactifications of microbundles are studied in Theorem 34 and Corollary 35.

All main results of this paper are obtained for the first time. They can be used for further studies not only in topology and its applications such as geometry, algebraic topology, representation theory and mathematical analysis, but also in mathematical physics.

2 Microbundles.

1. Notation. Let $\bf F$ be a topological ring such that its topology $\tau_{\bf F}$ is neither discrete nor antidiscrete. Then $({\bf F},\tau_{\bf F})$ is called a proper topological ring. We consider a topological left module $X_{\bf F}$ over $\bf F$, or shortly $X$ if $\bf F$ is specified. The ring $\bf F$ is supposed to be associative and commutative relative to the addition, but may be noncommutative or nonassociative relative to the multiplication.

Henceforward, it will also be written shortly a ring or a module instead of a topological ring or a topological module. Their homomorphisms will be supposed being continuous. Neighborhoods in topological spaces, modules, rings will be open if something other will not be specified and the topological terminology is used in the sense of the book [3].

2. Definition. Suppose the following.

$(2.1)$ There are are topological spaces $A$ and $E$.

$(2.2)$ There are continuous maps $i:A\to E$ and $p:E\to A$ such that $p\circ i=id$, where $id:A\to A$ is the identity map, $p\circ i$ denotes a composition of maps. Then $A$ will be called a base space, $E$ will be called a total space.

$(2.3)$ For each $b\in A$ neighborhoods $U$ of $b$ and $V$ of $i(b)$ exist such that $i(U)\subset V$ and $p(V)\subset U$ and $V$ is homeomorphic to $U\times X$, where $h_{V}:V\to U\times X$ is a homeomorphism.

$(2.4)$ There are continuous maps: a projection $\hat{\pi}_{1}:U\times X\to U$, an injection $\iota_{0}:U\to U\times X$, also a projection $\hat{\pi}_{2}:U\times X\to X$ such that $\hat{\pi}_{1}(d,x)=d$, $~{}\iota_{0}(d)=(d,0)$ and $\hat{\pi}_{2}(d,x)=x$ for each $d\in U$ and $x\in X$. They are supposed to satisfy the identity: $\hat{\pi}_{1}\circ\iota_{0}|_{U}=p|_{V}\circ i|_{U}$, where $i|_{U}$ denotes the restriction of $i$ to $U$.

If the conditions $(2.1)-(2.4)$ are satisfied, then it will be said that they define a microbundle ${\cal B}={\cal B}(A,E,{\bf F},X,i,p)$ with a fibre $X=X_{\bf F}$ of ${\cal B}$.

If a fibre $X$ is finite dimensional over the ring $\bf F$, that is $X={\bf F}^{n}$ with $n\in\bf N$, then $n$ is called the fibre dimension of ${\cal B}$ over ${\bf F}$. If $X$ is infinite dimensional over ${\bf F}$, then it is said that the microbundle $\cal B$ has an infinite fibre dimension over ${\bf F}$. If some data are specified, like $\bf F$ or $X$, they can be omitted from ${\cal B}(A,E,{\bf F},X,i,p)$ in order to shorten the notation.

Examples. 3.1. In particular, if $E=A\times X~{}$, $i=\iota_{0}~{}$, $p=\hat{\pi}_{1}$, then such a microbundle will be called the standard trivial microbundle and it will be denoted by ${\sf s}_{A,X}$ or ${\sf s}$.

3.2. Suppose that $\xi$ is a vector bundle over $A$ with a fibre $X$ over a field $\bf F$ and a structure group $GL(X)$ of all continuous linear automorphisms $T:X\to X$. Suppose also that $E$ is its total space, $p:E\to A$ is a projection, $i:A\to E$ is a zero cross-section. This provides an underlying microbundle $|\xi|$ of $\xi$.

4. Definitions. Let ${\cal B}_{1}={\cal B}(A_{1},E_{1},X_{1},i_{1},p_{1})$ and ${\cal B}_{2}={\cal B}(A_{2},E_{2},X_{2},i_{2},p_{2})$ be two microbundles. Let also neighborhoods $V_{1}$ of $i_{1}(A_{1})$ in $E_{1}$ and $V_{2}$ of $i_{2}(A_{2})$ in $E_{2}$ and homeomorphisms $g:V_{1}\to V_{2}$ and $s:A_{1}\to A_{2}$ exist such that $p_{2}\circ g|_{V_{1}}=s\circ{p_{1}}|_{V_{1}}$ and $g\circ i_{1}=i_{2}\circ s$ and $p_{2}\circ i_{2}\circ s=s\circ p_{1}\circ i_{1}$. Then these microbundles are called base neighbor isomorphic. For short it will be said ”isomorphic microbundles” instead of ”base neighbor isomorphic microbundles”.

A microbundle is called trivial if it is isomorphic to the standard trivial microbundle ${\sf s}$. We remind also the following.

Let $M$ be a topological space and let $X$ be a left module over a proper topological ring $\bf F$. Suppose also that

$(4.1)$ $M$ has a covering $\{U_{k}:~{}k\in K\}$, that is $\bigcup_{k\in K}U_{k}=M$, where $U_{k}$ is an open subset in $M$ for each $k\in K$, where $K$ is a set;

$(4.2)$ for each $k\in K$ there exists a homeomorphism $\phi_{k}:U_{k}\to V_{k}$, where $V_{k}$ is open in $X$.

Then $M$ is called a topological manifold on $X$. A triple $(U_{k},\phi_{k},V_{k})$ is called a chart, where $k\in K$. A collection $\{(U_{k},\phi_{k},V_{k}):k\in K\}$ of charts is called an atlas of $M$ and denoted by $At(M)$.

5. Lemma. Suppose that $M$ is a topological manifold on $X$ possessing an atlas of $M$ with charts homeomorphic to $X$ and supplied with a diagonal map $\Delta:M\to M\times M$. Then this manifold induces a microbundle with $A=M$, $E=M\times M$, $i=\Delta$.

Proof. Evidently $\hat{\pi}_{1}\circ\Delta=id$ on $M$. For any $m\in M$ take a neighborhood $U$ such that a homeomorphism $f:U\to X$ exists. It induces a map $g:U\times U\to U\times X$ such that $g(x,y)=(x,f(y)-f(x))$. Then $g$ is a homeomorphism of $U\times U$ onto $U\times X$ such that $\hat{\pi}_{1}\circ\Delta|_{U}=\hat{\pi}_{1}\circ g$ and $g\circ\Delta|_{U}={\iota_{0}}|_{U}$ and ${\hat{\pi}_{1}}|_{U\times U}=\hat{\pi}_{1}\circ g$.

6. Definition. The microbundle provided by Lemma 5 is called the tangent microbundle of $M$ and denoted by ${\sf t}_{M}$ or ${\sf t}$.

7. Note. We consider a local field $\bf K$. This is a finite algebraic extension of the field $\bf Q_{p}$ of $p$-adic numbers with a multiplicative nontrivial norm $|\cdot|_{\bf K}$ extending that of $\bf Q_{p}$ [18, 19].

Using antiderivation operators in the sense of Schikhof [14] in Section 2 in [8] and in Section 3 in [10] were defined and investigated manifolds over $\bf K$ of classes $\mbox{ }_{P}C_{0}((t,s))$ and $\mbox{ }_{S}^{l}C^{(q+l,n-1)}$ respectively.

It appears that for them tangent microbundle structures also exist.

8. Theorem. Let $M$ be over ${\bf F}=\bf K$ either a $\mbox{ }_{P}C_{0}((t,s))$-manifold with $s\geq 2$ and $t\geq 0$ or a $\mbox{ }_{S}^{l}C^{(q+l,n-1)}$-manifold with $l\geq 2$, $q\geq 0$ and $n\geq 1$ (see Note 7). Let also $TM$ be its tangent vector bundle. Then the underlying microbundle $|TM|$ corresponding to $TM$ is isomorphic to the tangent microbundle ${\sf t}_{M}$ of $M$.

Proof. In the first case in view of Theorem 2.7 in [8] a clopen neighborhood ${\tilde{T}}M$ of $M$ in $TM$ exists together with an exponential $C_{0}((t,s))$-mapping $\exp:{\tilde{T}}M\to M$ of ${\tilde{T}}M$ on $M$. In the second case by virtue of Theorem 3.23 in [10] there exist a clopen neighborhood ${\tilde{T}}M$ of $M$ in $TM$ and an exponential $\mbox{ }_{S}^{l}C^{(q+l,n-1)}$-mapping $\exp:{\tilde{T}}M\to M$ of ${\tilde{T}}M$ on $M$. This mapping $\exp$ is induced by considering geodesics in $M$ over $\bf K$.

There exists the natural embedding $\phi:M\to{\tilde{T}}M$ such that $M\ni y\mapsto\phi(y)=(x,0)\in{\tilde{T}}M$. Therefore, a map $f:{\tilde{T}}M\to M\times M$ can be defined for which

$f(x,v)=(x,\exp_{x}(v))$ for each $(x,0)\in\phi(M)$ and $v\in{\tilde{T}}_{x}M$. We apply to $f$ a non-archimedean analog of the Taylor Theorem A.1 in [11] and the inverse function Theorem A.4 in [7, 9] (see also [14]). They imply that for each $(x,0)\in\phi(M)$ a neighborhood $U_{x}$ of $(x,0)$ in ${\tilde{T}}M$ exists such that the restriction $f|_{U_{x}}$ is a diffeomorphism on a neighborhood $V_{y}$ of $(y,y)\in M\times M$, where $y\in M$ and $\phi(y)=(x,0)$. Taking a covering of the diagonal $D_{M}=\{(y,y):y\in M\}$ in $M\times M$ by such neighborhoods $V_{y}$ provides neighborhoods $V$ of $D_{M}$ in $M\times M$ and $U$ of $\phi(M)$ in ${\tilde{T}}M$ such that $f:U\to V$ is a diffeomorphism of $U$ onto $V$. Therefore, from the commutative diagram of this situation it follows that the underlying microbundle $|TM|$ corresponding to $TM$ is isomorphic to the tangent microbundle ${\sf t}_{M}$ of $M$.

9. Definition. Let $\bf F$ be a unital topological ring.

A topological space $A$ will be called $\bf F$ completely regular, if it is $T_{1}$ and for each closed subset $V$ in $A$ and each point $a\in A\setminus V$ a continuous function $f:A\to\bf F$ exists such that $f(a)=0$ and $f(V)=\{1\}$.

10. Realization of trivial microbundles.

Let $\bf F$ be an infinite field with

$(10.1)$ a topology induced by a multiplicative norm $|\cdot|_{\bf F}$, where the norm takes values in $[0,\infty)=\{t\in{\bf R}:~{}t\geq 0\}$ and

$(10.2)$ let $\Gamma_{\bf F}$ be dense in $(0,\infty)$,

where $\Gamma_{\bf F}=\{|b|_{\bf F}:0\neq b\in{\bf F}\}$; let also

$(10.3)$ let also $\bf F$ be of zero characteristic $char({\bf F})=0$.

Suppose that $X$ is a Banach space over the field $\bf F$ with a nontrivial norm taking values in $\Gamma_{\bf F}\cup\{0\}$. Suppose also that ${\sf s}_{A,X}$ is a trivial microbundle with a paracompact $\bf F$ completely regular base space $A$ of zero dimension $dim(A)=0$ and a fibre $X$

11. Proposition. If conditions of subsection 10 are satisfied, then an open subset $U_{0}$ in ${\sf s}_{A,X}$ exists such that it is homeomorphic to $A\times X$. Moreover, this homeomorphism is compatible with the injection and projection maps.

Proof. Using the definition of the trivial microbundle ${\sf s}_{A,X}$ we consider an open subset $U$ of $A\times X$.

For each closed subset $V$ in $A$ and each point $a\in A\setminus V$ open neighborhoods $W_{a}$ of $a$ and $W_{V}$ of $V$ exist which are disjoint $W_{a}\cap W_{V}=\emptyset$, since a continuous function $f:A\to\bf F$ exists such that $f(a)=0$ and $f(V)=\{1\}$. Indeed, one can take $W_{a}=f^{-1}(\{b\in{\bf F}:|b|<r_{1}\})$ and $W_{V}=f^{-1}(\{b\in{\bf F}:r_{2}<|b|\})$, where $0<r_{1}<r_{2}<1$, $r_{1}$ and $r_{2}$ belong to $\Gamma_{\bf F}$. An existence of such $r_{1}$ and $r_{2}$ follows from the condition $(10.2)$, since $|0|_{\bf F}=0$ and $|1|_{\bf F}=1$ and $|b|_{\bf F}>0$ for each $b\neq 0$. Thus $A$ is a $T_{3}$ space.

For each $a\in A$ a radius $0<r(a)<\infty$ with $r(a)\in\Gamma_{\bf F}$ exists such that $(a,x)\in U$ for each $x\in X$ with $|x|_{X}<r(a)$. Therefore using the base of the topology in the product $A\times X$ we infer that an open neighborhood $W_{a}$ of $a$ exists such that $\rho(a):=\inf\{r(b):b\in W_{a}\}>0$, since $U$ is open in $A\times X$. Thus a covering ${\cal V}=\{V_{a}:~{}a\in A\}$ of $U_{1}$ exists with $V_{a}=\{(b,x)\in U:b\in W_{a},x\in X,|x|_{X}<r(b)\}$, where $U_{1}=\bigcup_{a\in A}V_{a}$ is a proper open subset in $A\times X$. Since $A$ is paracompact, this covering ${\cal V}$ of $U_{1}$ contains a subcovering ${\cal W}\subset\cal V$ such that ${\cal W}=\{V_{a}:~{}a\in A_{0}\}$ with $A_{0}\subset A$ and $\{W_{a}:a\in A_{0}\}$ is a locally finite covering of $A$.

The topological space $A$ is normal, since it is $T_{1}\cap T_{3}$ and paracompact (see Section 1.5 and Theorem 5.1.5 in [3]). Let ${\cal P}=\{P_{j}:~{}j\in J\}$ be an open locally finite covering of $A$, where $J$ is some set, $A=\bigcup_{j\in J}P_{j}$. From the lemma about shrinking of covering (see Lemma 5.1.6 in [3]) it follows that it contains a covering ${\cal C}=\{C_{j}:~{}j\in J\}$ by closed subsets $C_{j}$ such that $C_{j}\subset P_{j}$ for each $j\in J$; $~{}A=\bigcup_{j\in J}C_{j}$. On the other hand, the topological space $A$ is zero-dimensional, consequently, each subset $C_{j}$ can be chosen clopen (closed and open simultaneously) in $A$ (see Sections 6.2 and 7.1 in [3]).

For each $C_{j}$ a continuous function $f_{j}:X\to{\bf F}$ exists such that $f_{j}(a)=0$ for each $a\in A\setminus C_{j}$ and $f_{j}(a)=1$ for each $a\in C_{j}$. Then we take a function $f(a)=\sum_{j\in J}f_{j}(a)$. Since $char({\bf F})=0$ and the covering $\cal C$ is locally finite, then $0<|f(a)|<\infty$ for each $a\in A$. This implies that a function $g_{j}(a)=f_{j}(a)/f(a)$ is continuous for each $j\in J$ and their sum $g(a)=\sum_{j\in J}g_{j}(a)=1$ is unit for each $a\in A$. Thus a family $\{g_{j}:j\in J\}$ is the partition of unity for the covering $\cal P$.

Then we consider balls $B({\bf F},t_{0},r)=\{t\in{\bf F}:|t-t_{0}|_{\bf F}\leq r\}$ in ${\bf F}$, where $t_{0}\in{\bf F}$, $~{}0<r<\infty$. Applying this partition of unity to $\cal W$ we get a continuous function $h:A\to(B({\bf F},0,1)\setminus\{0\})$ such that if $(a,x)\in A\times X$ and $|x|_{X}<|h(a)|_{\bf F}$, then $(a,x)\in U$, where $~{}|h(a)|_{\bf F}\geq\min(\rho(a),1)r_{2}$, where $r_{2}\in\Gamma_{\bf F}$ with $0<r_{2}<1$. For each $x\in X$ it is possible to choose $\xi(x)\in\bf F$ such that $|x|_{X}=|\xi(x)|_{\bf F}$, since $|x|_{X}\in\Gamma_{\bf F}\cup\{0\}$. We put $\psi(a,x)=(a,(h(a)-\xi(x))^{-1}x)$ for each $(a,x)\in U_{0}$, where $U_{0}=\{(a,x):a\in A,x\in X,|x|_{X}<|h(a)|_{\bf F}\}$.

Since $\Gamma_{\bf F}$ is dense in $(0,\infty)$, then for each $\epsilon>0$ and $a\in A$ a vector $x\in X$ exists such that $(a,x)\in U_{0}$ and $|h(a)-\xi(x)|_{\bf F}<\epsilon$. Therefore $\psi:U_{0}\to A\times X$ is a homeomorphism of an open proper subset $U_{0}$ of ${\sf s}_{A,X}$ onto ${\sf s}_{A,X}$.

12. Definition. For a microbundle ${\cal B}(A,E,{\bf F},X,i,p)$ and a topological space $A_{1}$ and a continuous map $f:A_{1}\to A$ an induced microbundle ${\cal B}(A_{1},E_{1},{\bf F},X,i_{1},p_{1})$ is defined with a total space $E_{1}=\{(a_{1},e)\in A_{1}\times E:f(a_{1})=p(e)\}$, where $i_{1}(a_{1})=(a_{1},i(f(a_{1})))$ for each $a_{1}\in A_{1}$, $~{}p_{1}(a_{1},e)=a_{1}$ for each $(a_{1},e)\in E_{1}$. The induced microbundle ${\cal B}(A_{1},E_{1},{\bf F},X,i_{1},p_{1})$ is also denoted by $f^{*}{\cal B}(A,E,X,i,p)$.

Particularly if $f$ is an inclusion map of $A_{1}$ into $A$, then $f^{*}{\cal B}(A,E,{\bf F},X,i,p)$ is a so called restricted microbundle ${\cal B}(A,E,{\bf F},X,i,p)|_{A_{1}}={\cal B}(A_{1},E_{2},{\bf F},X,i_{2},p_{2})$ with $E_{2}=p^{-1}(A_{1})$, $~{}i_{2}=i|_{A_{1}}$, $~{}p_{2}=p|_{E_{2}}$.

13. Cone over $\bf F$.

Assume that

$(13.1)$ $\bf F$ is an infinite unital ring with a topology induced by a nontrivial norm taking values in $[0,\infty)$.

We put $CA_{1}=(A_{1}\times B({\bf F},0,1))/(A_{1}\times\{0\})$ to be a cone of a topological space $A_{1}$ over the ring $\bf F$, where $B({\bf F},x,r):=\{y\in{\bf F}:~{}|x-y|_{\bf F}\leq r\}$, $x\in\bf F$, $0<r<\infty$.

$(13.2)$ For topological spaces $A$ and $A_{1}$ and a continuous map $f:A_{1}\to A$

let $A\bigcup_{f}CA_{1}=(A\cup CA_{1})/\Xi$ be a mapping cone of $f$, where $\Xi=\Xi_{f}$ denotes an identification $(a_{1},1)\Xi f(a_{1})$ for each $a_{1}\in A_{1}$.

14. Lemma. If Conditions $(13.1)$ and $(13.2)$ are fulfilled and a ring $\bf F$ is path-connected and a microbundle ${\cal B}(A,E,{\bf F},X,i,p)$ can be extended to a microbundle over $A\bigcup_{f}CA_{1}$, then $f^{*}{\cal B}(A,E,{\bf F},X,i,p)$ is trivial.

Proof. Apparently the composition ${A_{1}}_{\overrightarrow{f}}A_{\overrightarrow{q}}A\bigcup_{f}CA_{1}$ is null-homotopic, since $\bf F$ is path-connected, where $q$ is an embedding of $A$ into $A\bigcup_{f}CA_{1}$. Consequently, $f^{*}{\cal B}(A,E,{\bf F},X,i,p)$ is trivial.

15. Lemma. If Conditions $(10.1)-(10.3)$ and $(13.2)$ are satisfied and an induced microbundle $f^{*}{\cal B}(A,E,{\bf F},X,i,p)$ is trivial, then ${\cal B}(A,E,{\bf F},X,i,p)$ can be extended over $A\bigcup_{f}CA_{1}$.

Proof. At first we take the mapping cylinder $Z=A\bigcup_{f}(A_{1}\times B({\bf F},0,1))$ of $f$, where $A\bigcup_{f}(A_{1}\times B({\bf F},0,1))=[A\bigcup(A_{1}\times B({\bf F},0,1))]/\Xi_{f}$. Then the microbundle ${\cal B}(A,E,{\bf F},X,i,p)$ can be extended to a microbundle ${\cal B}_{1}$ over $Z$, since $A$ is a retract of $Z$. Therefore the restriction ${\cal B}_{1}|_{A_{1}\times\{0\}}$ is trivial as well, consequently, ${\cal B}_{1}|_{A_{1}\times B({\bf F},0,r)}$ is trivial for each $0<r<1$ with $r\in\Gamma_{\bf F}$. We fix such $r$.

By virtue of Proposition 11 an open subset $U_{0}$ of the total space $E_{1,r}$ of the restricted microbundle ${\cal B}_{1}|_{A_{1}\times B({\bf F},0,r)}$ is homeomorphic to $A_{1}\times B({\bf F},0,r)\times X$ with a homeomorphism $h$ compatible with injections and projections. Since $A\bigcup_{f}CA_{1}=Z/(A_{1}\times\{0\})$, then it induces from ${\cal B}_{1}$ a microbundle ${\cal B}_{2}$ over $A\bigcup_{f}CA_{1}$. It remains to note that a total space $E_{2}$ of ${\cal B}_{2}$ is obtained from $E_{1}$ by an identification $h^{-1}(A_{1}\times\{0\}\times x)$ with $x$ for each $x\in X$, where $E_{1}$ is a total space of ${\cal B}_{1}$.

16. Proposition. If Conditions $(13.1)$ and $(13.2)$ are satisfied and a ring $\bf F$ is zero-dimensional, $dim({\bf F})=0$, then a microbundle ${\cal B}(A,E,{\bf F},X,i,p)$ can be extended to a microbundle over $A\bigcup_{f}CA_{1}$.

Proof. At first we take a partition of the unit ball $B({\bf F},0,1)$ into two disjoint clopen subsets $K_{0}$ and $K_{1}$ such that $0\in K_{0}$ and $1\in K_{1}$, that is $K_{0}\cap K_{1}=\emptyset$ and $K_{0}\cup K_{1}=B({\bf F},0,1)$, since $dim({\bf F})=0$. Therefore $A\bigcup_{f}CA_{1}$ is the disjoint union of two clopen subsets $A_{2}:=[A\bigcup(A_{1}\times K_{1})]/\Xi_{f}$ and $A_{3}:=[A\bigcup(A_{1}\times K_{0})]/(A_{1}\times\{0\})$.

Let $T:X\to X$ be any left $\bf F$ linear automorphism of a topological left module $X$ over $\bf F$ (that is $T$ and $T^{-1}$ are continuous), $V=V_{b}$ be a neighborhood of $b=i\circ f(a_{1})$ in $E$, $h_{V}:V\to U\times X$ be a homeomorphism, $U$ be a neighborhood of $f(a_{1})$ in $A$, $\hat{\pi}_{2}:U\times X\to X$ be a projection such that $\hat{\pi}_{2}(d,x)=x$ for each $d\in U$ and $x\in X$ (see Definition 2). We put $E_{2}=[E\bigcup(A_{1}\times K_{1}\times X)]/\Xi_{g}$, where $g:A_{1}\times X\to E$ is a continuous mapping such that $p\circ g(a_{1},x)=i\circ f(a_{1})$ and $\hat{\pi}_{2}\circ h_{V}\circ g(a_{1},x)=Tx$ for each $a_{1}\in A_{1}$ and $x\in X$, where $\Xi_{g}$ identifies $(a_{1},1,x)$ with $g(a_{1},x)$.

Take any automorphism $g_{2}:B({\bf F},0,1)\to B({\bf F},0,1)$ such that $g_{2}(1)=1$ and $g_{2}(0)=0$ (that is $g_{2}$ and $g^{-1}_{2}$ are continuous). An injection $i:A\to E$ has a continuous extension $i:A_{2}\to E_{2}$ such that $p\circ i(a_{1},t)=(i\circ f(a_{1}),g_{2}(t))$ and $\hat{\pi}_{2}\circ h_{V}\circ i(a_{1},t)=(t+\beta(1-t))\hat{\pi}_{2}\circ h_{V}\circ i\circ f(a_{1})$ for each $t\in K_{1}\setminus\{1\}$ and $a_{1}\in A_{1}$, where $V$ is a neighborhood of $i(a_{1},t)$ in $E_{2}$, $~{}\beta$ is a fixed element in $\bf F$. Therefore the projection $p:E\to A$ has a continuous extension on $E_{2}$ such that $p:E_{2}\to A_{2}$ with $p\circ h_{V}^{-1}(b,t,x)=(a_{1},g_{2}^{-1}(t))$ for each $t\in K_{1}\setminus\{1\}$ and $a_{1}\in A_{1}$ with $b=i\circ f(a_{1})$. That is $p\circ i=id$ on $A_{2}$. Thus a microbundle ${\cal B}_{2}={\cal B}(A_{2},E_{2},{\bf F},X,i,p)$ is an extension of ${\cal B}(A,E,{\bf F},X,i,p)$.

On $A_{3}$ a microbundle ${\cal B}_{3}={\cal B}(A_{3},E_{3},{\bf F},X,i_{3},p_{3})$ exists, which may be in particular trivial ${\sf s}_{A_{3},X}$.

For mappings $f_{j}:B_{j}\to C_{j}$ for each $j\in\{1,2\}$ and $B=B_{1}\cup B_{2}$ and $C=C_{1}\cup C_{2}$ with $B_{1}\cap B_{2}=\emptyset$ and $C_{1}\cap C_{2}=\emptyset$ by $f_{1}\nabla f_{2}$ is denoted their combination such that $(f_{1}\nabla f_{2})(b_{j})=f_{j}(b_{j})$ for each $b_{j}\in B_{j}$ and $j\in\{1,2\}$. Therefore the combination ${\cal B}_{2}\nabla{\cal B}_{3}={\cal B}(A_{2}\cup A_{3},E_{2}\cup E_{3},{\bf F},X,i_{2}\nabla i_{3},p_{2}\nabla p_{3})$ of microbundles ${\cal B}_{2}$ and ${\cal B}_{3}$ provides the extension over $A\bigcup_{f}CA_{1}$ of the microbundle ${\cal B}(A,E,{\bf F},X,i,p)$.

17. Definition. Suppose that $M_{1}$ and $M$ are topological manifolds (see Definition 4) on ${\bf F}^{m_{1}}$ and ${\bf F}^{m}$ over a topological ring $\bf F$ such that $M_{1}\subset M$, where $m_{1}$ and $m$ are cardinals such that $m_{1}\leq m$, where ${\bf F}^{m}$ is supplied with the Tychonoff product topology, ${\bf F}$ has the topological weight $\tau=w{\bf F}\geq\aleph_{0}$. Suppose also that a neighborhood $U$ of $M_{1}$ in $M$ exists and a retraction $p:U\to M_{1}$ is such that ${M_{1}}_{\overrightarrow{i}}U_{\overrightarrow{p}}M_{1}$ forms a microbundle, where $i$ is an inclusion map. Then it will be said that $M_{1}$ has a microbundle neighborhood in $M$. It will be denoted by ${\cal N}={\cal N}(M_{1},M,i,p)$.

In particular, if $U$ and $p$ can be chosen such that ${\cal N}$ is trivial, then $M_{1}$ has a product neighborhood.

18. Corollary. Assume that Conditions $(10.1)-(10.3)$ are satisfied. Then in the notation of Definition 17 $M_{1}$ has a trivial microbundle neighborhood if and only if a neighborhood $U$ of $M_{1}$ in $M$ exists such that the pair $(U,M_{1})$ is homeomorphic to $M_{1}\times(X,0)$.

This follows from Proposition 11.

19. Theorem. If $M_{1}$ and $M$ are topological manifolds over a topological ring $\bf F$ of zero small inductive dimension, and $M_{1}$ is a closed submanifold in $M$, then a retraction $\check{r}:M\to M_{1}$ exists.

Proof. Since $ind({\bf F})=0$, then the small inductive dimension of ${\bf F}^{m}$ is zero, $ind({\bf F}^{m})=0$ (see Theorem 6.2 and Section 7.1 in [3]). An atlas of $M$ has a refinement being an atlas with charts homeomorphic to clopen subsets in ${\bf F}^{m}$, since ${\bf F}^{m}$ has a base of its topology consisting of clopen subsets. Thus we consider that each chart $V_{j}$ of the atlas $At(M)=\{(V_{j},f_{j}):j\in J\}$ of $M$ is homeomorphic to a clopen subset $W_{j}=f_{j}(V_{j})$ in ${\bf F}^{m}$, where $J$ is a set. A similar choice of an atlas $At(M_{1})$ can be made for $M_{1}$.

Note that ${\bf F}^{m}$ has an embedding $g$ into the generalized Cantor discontinuum $D^{\tau m}$, because $\tau\geq\aleph_{0}$. Therefore, $M$ has an embedding into $D^{t}$, where $t=\tau mn$, $\tau$ is a topological weight of $\bf F$, $n$ is a cardinality of $J$. We take the closure $C=cl(g(M))$ of $g(M)$ in $D^{t}$. There exists a retraction $q:C\to C_{1}$, where $C_{1}=cl(g(M_{1}))$ (see [3, 4, 5]). Therefore, the restriction $q|_{g(M)}$ induces $\check{r}=g^{-1}\circ q\circ g|_{V}$, where $V=\check{r}^{-1}(M_{1})=M$, since $M_{1}$ is closed in $M$; $~{}g(M_{1})$ is dense in the compact space $C_{1}$ and $q(c_{1})=c_{1}$ for each $c_{1}\in C_{1}$.

20. Theorem. Assume that manifolds $M$ and $M_{1}$ are both over a field ${\bf F}=\bf K$ either $\mbox{ }_{P}C_{0}((t,s))$-manifolds with $s\geq 2$ and $t\geq 0$ or $\mbox{ }_{S}^{l}C^{(q+l,n-1)}$-manifolds with $l\geq 2$, $q\geq 0$ and $n\geq 1$ (see Definitions 6, 17 and Note 7), where $M_{1}$ is a closed submanifold in $M$. Then the total space of $i^{*}{\sf t}_{M}$ is homeomorphic to the total space of $\check{r}^{*}{\sf t}_{M_{1}}$.

Proof. By virtue of Theorem 19 we consider the case when there is a retraction $\check{r}:M\to M_{1}$. The total space $E=E(i^{*}{\sf t}_{M})$ of $i^{*}{\sf t}_{M}$ consists of all pairs $(m_{1},(m,k))\in M_{1}\times M^{2}$ such that $i(m_{1})=m$, consequently, it is homeomorphic to $M_{1}\times M$. Then $E_{1}=E(\check{r}^{*}{\sf t}_{M_{1}})$ is a subspace in $M\times M_{1}^{2}$ composed of all pairs $(m,(m_{1},k))$ with $\check{r}(m)=m_{1}$, hence the total space $E_{1}$ is homeomorphic to $M\times M_{1}$.

21. Corollary. If the conditions of Theorem 20 are accomplished, then the submanifold $M_{1}$ has a microbundle neighborhood ${\cal N}$ homeomorphic with $i^{*}{\sf t}_{M}$.

22. Definition. If a microbundle ${\sf t}_{M}$ is trivial, then a manifold $M$ is called topologically parallelizable.

23. Corollary. If conditions of Theorem 20 are fulfilled and $M_{1}$ is topologically parallelizable, then $M_{1}\times\{0\}$ has a microbundle neighborhood in $M\times X$ with a normal microbundle ${\cal N}$ homeomorphic to $i^{*}{\sf t}_{M}$.

Proof. Since ${\sf t}_{M_{1}}$ is trivial, then $\check{r}^{*}{\sf t}_{M_{1}}$ is trivial as well. Therefore, the total space $E(\check{r}^{*}{\sf t}_{M_{1}})$ is homeomorphic with $M\times X$ of the canonical trivial microbundle ${\sf s}_{M,X}$, where $X={\bf F}^{m}$.

24. Corollary. Let the conditions of Theorem 20 be satisfied and let $M$ and $M_{1}$ be topologically parallelizable. Then $M_{1}\times\{0\}$ has a product neighborhood $V$ in $M\times X$.

Proof. Since ${\sf t}_{M}$ is trivial, then ${\cal N}$ is trivial.

25. Corollary. Let $M$ be a compact topologically parallelizable of dimension $1\leq m=dim_{\bf K}M<\infty$ over a field ${\bf F}=\bf K$ either $\mbox{ }_{P}C^{(q,n)}$-manifolds or $\mbox{ }_{S}^{l}C^{(q,n)}$-manifolds with $q\geq 1$ and $n\geq 0$ (see Definitions 6 and 17).

Then a product neighborhood $V$ of $M\times{\bf K}^{2m+1}$ exists such that it can be embedded into ${\bf K}^{3m+1}$ as a clopen subset.

Proof. By virtue of Theorem 3.21 in [10] there exists a $\mbox{ }_{S}C^{(q,n)}$ or $\mbox{ }_{P}C^{(q,n)}$-embedding $\tau:M\hookrightarrow{\bf K}^{2m+1}$ correspondingly. Since $\bf K$ is the local field, it is zero-dimensional $dim({\bf K})=0$ (see [3, 13, 18]). Each ball $B({\bf K}^{n},x,r)=\{y\in{\bf K}^{n}:|x-y|_{\bf K}\leq r\}$ is clopen in ${\bf K}^{n}$, where $x\in{\bf K}^{n}$, $r\in\Gamma_{\bf K}$, $n\in{\bf N}$. Therefore a product neighborhood $V$ provided by Corollary 24 can be chosen homeomorphic to a clopen subset in ${\bf K}^{3m+1}$.

26. Definition. Let ${\cal B}_{k}={\cal B}(A_{k},E_{k},{\bf F}_{k},X_{k},i_{k},p_{k})$ be a family of microbundles over topological rings ${\bf F}_{k}$ with $k\in\Lambda$, where $\Lambda$ is a directed set. Let also for each $k\leq n$ in $\Lambda$ a homomorphism $\pi^{n}_{k}:{\cal B}_{n}\to{\cal B}_{k}$ be given satisfying the conditions $(26.1)-(26.5)$:

$(26.1)$ $\pi^{n}_{k}=(\pi^{1,n}_{1,k};\pi^{2,n}_{2,k};\pi^{3,n}_{3,k};\pi^{4,n}_{4,k})$ with

$(26.2)$ $\pi^{1,n}_{1,k}:A_{n}\to A_{k}$ and $\pi^{2,n}_{2,k}:E_{n}\to E_{k}$,

$\pi^{3,n}_{3,k}:{\bf F}_{n}\to{\bf F}_{k}$ and $\pi^{4,n}_{4,k}:X_{n}\to X_{k}$, where

$(26.3)$ $\pi^{2,n}_{2,k}\circ i_{n}=i_{k}\circ\pi^{1,n}_{1,k}$ and $p_{k}\circ\pi^{2,n}_{2,k}=\pi^{1,n}_{1,k}\circ p_{n}$,

$(26.4)$ $h_{k}\circ\pi^{2,n}_{2,k}\circ h_{n}^{-1}(a_{n},u_{n}x_{n}+v_{n}y_{n})=$

$\pi^{3,n}_{3,k}(u_{n})h_{k}\circ\pi^{2,n}_{2,k}\circ h_{n}^{-1}(a_{n},x_{n})+\pi^{3,n}_{3,k}(v_{n})h_{k}\circ\pi^{2,n}_{2,k}\circ h_{n}^{-1}(a_{n},y_{n})$,

$(26.5)$ $\hat{\pi}_{2,k}\circ h_{k}\circ\pi^{2,n}_{2,k}=\pi^{4,n}_{4,k}\circ\hat{\pi}_{2,n}\circ h_{n}$ and

$\pi^{4,n}_{4,k}(u_{n}x_{n}+v_{n}y_{n})=\pi^{3,n}_{3,k}(u_{n})\pi^{4,n}_{4,k}(x_{n})+\pi^{3,n}_{3,k}(v_{n})\pi^{4,n}_{4,k}(y_{n})$

for every $u_{n}$ and $v_{n}$ in ${\bf F}_{n}$, $x_{n}$ and $y_{n}$ in $X_{n}$, $a_{n}\in A_{n}$, where $h_{n}:V_{n}\to U_{n}\times X_{n}$ is a local homeomorphism for an open subset $V_{n}$ in $E_{n}$ corresponding to an open neighborhood $U_{n}$ of a point $a_{n}$ in $A_{n}$; where $\pi^{n}_{n}$ is the identity homomorphism if $n=k$ and $\pi^{k}_{l}\circ\pi^{n}_{k}=\pi^{n}_{l}$ for each $l\leq k\leq n$ in $\Lambda$.

Such a family $\{{\cal B}_{n},\pi^{n}_{k},\Lambda\}$ will be called an inverse spectrum of microbundles.

Let $A=\lim\{A_{n},\pi^{1,n}_{1,k},\Lambda\}$ and $E=\lim\{E_{n},\pi^{2,n}_{2,k},\Lambda\}$ be limits of inverse spectra of topological spaces. Let also ${\bf F}=\lim\{{\bf F}_{n},\pi^{3,n}_{3,k},\Lambda\}$ and $X=\lim\{X_{n},\pi^{4,n}_{4,k},\Lambda\}$ be limits of inverse spectra of topological rings and topological left modules respectively.

Let ${\cal B}={\cal B}(A,E,{\bf F},X,i,p)$ be a microbundle such that for each $n\in\Lambda$ a homomorphism $\pi_{n}:{\cal B}\to{\cal B}_{n}$ exists satisfying analogous to $(26.1)$-$(26.5)$ conditions and the following condition: $\pi^{n}_{k}\circ\pi_{n}=\pi_{k}$ for each $k\leq n$ in $\Lambda$. Then it will be said that ${\cal B}$ is a limit of the inverse spectrum $\{{\cal B}_{n},\pi^{n}_{k},\Lambda\}$ of microbundles.

27. Theorem. If $\{{\cal B}_{n},\pi^{n}_{k},\Lambda\}$ is an inverse spectrum of microbundles, then its limit exists.

Proof. The inverse spectrum of microbundles induces inverse spectra of topological spaces $\{A_{n},\pi^{1,n}_{1,k},\Lambda\}$ and $\{E_{n},\pi^{2,n}_{2,k},\Lambda\}$. Therefore there exist topological spaces $A$ and $E$ being their limits $A=\lim\{A_{n},\pi^{1,n}_{1,k},\Lambda\}$ and $E=\lim\{E_{n},\pi^{2,n}_{2,k},\Lambda\}$ (see Section 2.5 in [3] and [16]). For each $k\in\Lambda$ there are projections $\pi_{1,k}$ from $A$ onto $A_{k}$ and $\pi_{2,k}$ from $E$ onto $E_{k}$.

Then from $(26.3)-(26.5)$ it follows that local homeomorphisms $h_{n}:V_{n}\to U_{n}\times X_{n}$ are compatible with inverse spectra of rings $\{{\bf F}_{n},\pi^{3,n}_{3,k},\Lambda\}$ and of left modules $\{X_{n},\pi^{4,n}_{4,k},\Lambda\}$, where $\pi^{3,n}_{3,k}:{\bf F}_{n}\to{\bf F}_{k}$ is a homomorphism of topological rings and $\pi^{4,n}_{4,k}:X_{n}\to X_{k}$ is a homomorphism of left topological modules such that

$(27.1)$ $\pi^{4,n}_{4,k}(u_{n}x_{n}+v_{n}y_{n})=\pi^{3,n}_{3,k}(u_{n})\pi^{4,n}_{4,k}(x_{n})+\pi^{3,n}_{3,k}(v_{n})\pi^{4,n}_{4,k}(y_{n})$

for every $u_{n}$ and $v_{n}$ in ${\bf F}_{n}$, $x_{n}$ and $y_{n}$ in $X_{n}$, $~{}k\leq n$ in $\Lambda$. For each $n\in\bf N$ the left module $X_{n}$ has also a structure of a commutative group relative to the addition on it.

Therefore there exist a topological ring ${\bf F}=\lim\{{\bf F}_{n},\pi^{3,n}_{3,k},\Lambda\}$ and a commutative group (relative to the addition) $X=\lim\{X_{n},\pi^{4,n}_{4,k},\Lambda\}$ (see [2, 3]). There are projections (homomorphisms) $\pi_{3,k}$ from ${\bf F}$ onto ${\bf F}_{k}$ and $\pi_{4,k}$ from $X$ onto $X_{k}$. Each $u$ in $\bf F$ has the form $u=(u_{n}:n\in\Lambda)$ such that $(\forall n\in\Lambda,u_{n}\in{\bf F}_{n},~{}\forall k\in\Lambda,\forall n\in\Lambda,~{}[(k\leq n)\Rightarrow(\pi^{3,n}_{3,k}(u_{n})=u_{k})])$. Each $x\in X$ is of the form $x=(x_{n}:n\in\Lambda)$ such that $(\forall n\in\Lambda,x_{n}\in X_{n},~{}\forall k\in\Lambda,\forall n\in\Lambda,~{}[(k\leq n)\Rightarrow(\pi^{4,n}_{4,k}(x_{n})=x_{k})])$. A base of a topology on $\bf F$ consists of all subsets $\pi_{3,k}^{-1}(S_{k})$ with $S_{k}$ open in ${\bf F}_{k}$ and $k\in\Lambda$. Similarly a base of a topology on $X$ is composed of all subsets $\pi_{4,k}^{-1}(Y_{k})$ with $Y_{k}$ open in $X_{k}$ and $k\in\Lambda$. Therefore ${\bf F}$ acts continuously on $X$ as $ux=(u_{n}x_{n}:n\in\Lambda)$ for each $u\in\bf F$ and $x\in X$, hence $X$ is a topological left module over $\bf F$.

We have that $\pi^{1,n}_{1,k}\circ\pi_{1,n}(a)=\pi_{1,k}(a)$ for each $a\in A$ and $\pi^{2,n}_{2,k}\circ\pi_{2,n}(e)=\pi_{2,k}(e)$ for each $e\in E$ and every $k\leq n$ in $\Lambda$. Therefore from Conditions $(26.1)$ and $(26.2)$ we infer that there exists an injection $i:A\to E$ such that $i(a)=(b_{n}:n\in\Lambda)$ satisfying $(\forall n\in\Lambda,b_{n}\in E_{n},~{}b_{n}=i_{n}(a_{n}))$ for each $a\in A$, since $a=(a_{n}:n\in\Lambda)$ such that $(\forall n\in\Lambda,~{}a_{n}\in A_{n},~{}\forall k\in\Lambda,~{}\forall n\in\Lambda,~{}[(k\leq n)\Rightarrow(\pi^{1,n}_{1,k}(a_{n})=a_{k})])$. Moreover, there exists a projection $p:E\to A$ such that $p(b)=(a_{n}:n\in\Lambda)$ satisfying the following condition $(\forall n\in\Lambda,a_{n}\in E_{n},~{}a_{n}=p_{n}(b_{n}))$ for each $b\in E$, since $b=(b_{n}:n\in\Lambda)$ such that $(\forall n\in\Lambda,~{}b_{n}\in E_{n},~{}\forall k\in\Lambda,~{}\forall n\in\Lambda,~{}[(k\leq n)\Rightarrow(\pi^{2,n}_{2,k}(b_{n})=b_{k})])$. Since $p_{n}\circ i_{n}=id_{n}$ for each $n\in\Lambda$, then $p\circ i=id$.

On the other hand, if $h_{k}:V_{k}\to U_{k}\times X_{k}$ is a homeomorphism, where $V_{k}$ is an open subset in $E_{k}$ and $U_{k}$ is an open subset in $A_{k}$, then $\pi_{2,k}^{-1}(V_{k})=V$ is open in $E$ and $\pi_{1,k}^{-1}(U_{k})=U$ is open in $A$; $~{}\pi_{4,k}(X)=X_{k}$; $~{}\pi_{3,k}({\bf F})={\bf F}_{k}$. Thus bases of topologies in $A$ and $E$ induce a local homeomorphism $h:V\to U\times X$ for the corresponding open subsets $V$ in $E$ and $U$ in $A$, where $h(v)=(a,x)$ with $(a,x)=((a_{n},x_{n}):n\in\Lambda)$ such that $(\forall n\in\Lambda,(a_{n},x_{n})=h_{n}(v_{n}),a_{n}\in A_{n},x_{n}\in X_{n})$ for each $v\in V$, where $v=(v_{n}:n\in\Lambda)$ such that $(\forall n\in\Lambda,v_{n}\in E_{n},~{}v_{n}=\pi_{2,n}(v))$. Hence $\hat{\pi}_{1}\circ h(v)=p(v)$ for each $v\in V$, since Identity $(27.1)$ is satisfied and $\hat{\pi}_{1,k}\circ h_{k}|_{V_{k}}=p_{k}|_{V_{k}}$ for each $k\in\Lambda$ (see Definition 2). There is the natural injection $\iota_{0}:U\hookrightarrow U\times X$. Then we deduce that $\hat{\pi}_{1}\circ\iota_{0}(a)=p|_{V}\circ i(a)$ for each $a\in U$, since $\hat{\pi}_{1,k}\circ\iota_{0,k}|_{U_{k}}=p_{k}|_{V_{k}}\circ i_{k}|_{U_{k}}$ for each $k\in\Lambda$. Thus ${\cal B}(A,E,{\bf F},X,i,p)=\lim\{{\cal B}_{n},\pi^{n}_{k},\Lambda\}$.

28. Definition. Let ${\bf S}_{1}=\{{\cal B}_{n},\pi^{n}_{k},\Lambda\}$ and ${\bf S}_{2}=\{{\cal C}_{n},\breve{\pi}^{n}_{k},\Upsilon\}$ be two inverse spectra of micronbundles, let also

$(28.1)$ $q:\Upsilon\to\Lambda$ be a map and

$(28.2)$ $T=\{\forall k\in\Upsilon~{}t_{k}:{\cal B}_{q(k)}\to{\cal C}_{k}\}$

be a family of homomorphisms satisfying analogous to $(26.1)$-$(26.5)$ conditions such that for each $k\leq n$ in $\Upsilon$ there exists $m\in\Lambda$ with $m\geq q(n)$ and $m\geq q(k)$ for which the following identity is satisfied:

$(28.3)$ $t_{k}\circ\pi^{m}_{q(k)}=\breve{\pi}^{n}_{k}\circ t_{n}\circ\pi^{m}_{q(n)}$.

Then $(q,T)$ is called a homomorphism of $\{{\cal B}_{n},\pi^{n}_{k},\Lambda\}$ into $\{{\cal C}_{n},\breve{\pi}^{n}_{k},\Upsilon\}$.

29. Theorem. There exists a covariant functor from a category of inverse spectra of microbunles ${\cal SB}$ into a category of microbundles ${\cal CB}$ induced by the operation $\lim$.

Proof. Let $(q,T)$ be a morphism of an inverse spectrum of microbundles ${\bf S}_{1}=\{{\cal B}_{n},\pi^{n}_{k},\Lambda\}$ into ${\bf S}_{2}=\{{\cal C}_{n},\breve{\pi}^{n}_{k},\Upsilon\}$, where ${\cal B}_{n}={\cal B}(A_{n},E_{n},{\bf F}_{n},X_{n},i_{n},p_{n})$ with a left module $X_{n}$ over a ring ${\bf F}_{n}$ for each $n\in\Lambda$; ${\cal C}_{n}={\cal B}(C_{n},D_{n},{\bf G}_{n},Y_{n},\breve{i}_{n},\breve{p}_{n})$ with a left module $Y_{n}$ over a ring ${\bf G}_{n}$ for each $n\in\Upsilon$. In view of Theorem 27 there exist limits ${\cal B}(A,E,{\bf F},X,i,p)=\lim{\bf S}_{1}$ and ${\cal B}(C,D,{\bf G},Y,\breve{i},\breve{p})=\lim{\bf S}_{2}$ of the inverse spectra of microbundles. Put

$(29.1)$ $(c_{k},d_{k},v_{k},y_{k})=t_{k}(a_{q(k)},b_{q(k)},u_{q(k)},x_{q(k)})$

for each $k\in\Upsilon$, where $(a,b,u,x)=((a_{k},b_{k},u_{k},x_{k}):k\in\Lambda)$ such that

$(\forall k\in\Lambda,~{}a_{k}\in A_{k},~{}b_{k}\in E_{k},~{}u_{k}\in{\bf F}_{k},~{}x_{k}\in X_{k})$.

For each $k\leq n$ in $\Upsilon$ there exists $m\in\Upsilon$ such that $m\geq k$ and $m\geq n$, since a set $\Upsilon$ is directed. From $(28.3)$ and $(26.2)$ it follows that

$(29.2)$ $(c_{k},d_{k},v_{k},y_{k})=\breve{\pi}^{n}_{k}(c_{n},d_{n},v_{n},y_{n})$.

Therefore $(29.1)$ and $(29.2)$ imply that a limit map $t=\lim(q,T)$ exists from ${\cal B}(A,E,{\bf F},X,i,p)$ into ${\cal B}(C,D,{\bf G},Y,\breve{i},\breve{p})$ such that

$(29.3)$ $t=(t^{1},t^{2},t^{3},t^{4})$ with

$(29.4)$ $t^{1}:A\to C$, $~{}t^{2}:E\to D$, $~{}t^{3}:{\bf F}\to{\bf G}$, $~{}t^{4}:X\to Y$

where ${\bf G}=\lim\{{\bf G}_{n},\breve{\pi}^{3,n}_{3,k},\Upsilon\}$ and $Y=\lim\{Y_{n},\breve{\pi}^{4,n}_{4,k},\Upsilon\}$. Then $(28.3)$ and $(26.3)$ imply that

$(29.5)$ $t^{2}\circ i=\breve{i}\circ t^{1}$ and $\breve{p}\circ t^{2}=t^{1}\circ p$.

From the construction of a topology on a limit of an inverse spectrum (see Subsection 27) it follows that $t$ is continuous. Therefore, applying $(28.3)$, $(26.4)$ and $(26.5)$ we deduce that

$(29.6)$ $\breve{h}\circ t^{2}\circ h^{-1}(a,ux+wy)=$

$t^{3}(u)\breve{h}\circ t^{2}\circ h^{-1}(a,x)+t^{3}(w)\breve{h}\circ t^{2}\circ h^{-1}(a,y)$ and

$(29.7)$ $\breve{\hat{\pi}}_{2}\circ\breve{h}\circ t^{2}=t^{4}\circ\hat{\pi}_{2}\circ h$ and $t^{4}(ux+wy)=t^{3}(u)t^{4}(x)+t^{3}(w)t^{4}(y)$

for every $u$ and $w$ in ${\bf F}$, $x$ and $y$ in $X$, $a\in A$, where $h:V\to U\times X$ is a local homeomorphism for an open subset $V$ in $E$ corresponding to an open neighborhood $U$ of a point $a$ in $A$. Therefore $t$ is a continuous homomorphism of microbundles, since it satisfies the conditions $(29.3)$-$(29.7)$. Note that $t$ is unique, since $\breve{\pi}_{n}\circ t=t_{n}\circ\pi_{q(n)}$ for each $n\in\Upsilon$, where $\pi_{n}:{\cal B}(A,E,{\bf F},X,i,p)\to{\cal B}_{n}$ with $\pi_{n}=(\pi_{1,n},\pi_{2,n},\pi_{3,n},\pi_{4,n})$ for each $n\in\Lambda$ (see Subsection 27).

It can be easily verified that a composition $(q_{1}\circ q_{2},T_{2}\circ T_{1})$ of morphisms $(q_{1},T_{1}):{\bf S}_{1}\to{\bf S}_{2}$ and $(q_{2},T_{2}):{\bf S}_{2}\to{\bf S}_{3}$ of inverse spectra ${\bf S}_{j}=\{{\cal B}_{j,k},~{}\mbox{}^{j}\pi^{n}_{k}:k\in\Lambda_{j}\}$ with $j\in\{1,2,3\}$ of microbundles ${\cal B}_{j,k}$ is a morphism from ${\bf S}_{1}$ into ${\bf S}_{3}$, where $T_{2}\circ T_{1}=\{\forall k\in\Lambda_{3},~{}t_{2,k}\circ t_{1,q_{2}(k)}:{\cal B}_{1,q_{1}(q_{2}(k))}\to{\cal B}_{3,k}\}$; $\Lambda_{j}$ is a directed set for each $j$. Thus the operation of taking the limit of an inverse spectrum of microbundles induces a covariant functor $\lim:{\cal SB}\to{\cal CB}$.

30. Theorem. Assume that ${\bf S}=\{{\cal B}_{n},\pi^{n}_{k},\Lambda\}$ is an inverse spectrum of microbundles and there are homomorphisms $t_{k}$ of a microbundle ${\cal C}={\cal B}(C,D,{\bf G},Y,\breve{i},\breve{p})$ into ${\cal B}_{k}$ such that $t_{k}=\pi^{n}_{k}\circ t_{n}$ for each $k\leq n$ in $\Lambda$. Then there exists a limit homomorphism $t=\lim\{t_{k}:k\in\Lambda\}$, $t:{\cal C}\to{\cal B}(A,E,{\bf F},X,i,p)$, such that $t_{k}=\pi_{k}\circ t$ for each $k\in\Lambda$, where ${\cal B}(A,E,{\bf F},X,i,p)=\lim\{{\cal B}_{n},\pi^{n}_{k},\Lambda\}$. Moreover, if $t_{k}({\cal C})$ is dense in $X_{k}$ for each $k\in\Lambda$, then $t({\cal C})$ is dense in ${\cal B}(A,E,{\bf F},X,i,p)$.

Proof. By virtue of Theorem 29 a continuous homomorphism $t$ of microbundles exists, since ${\cal C}$ can be written as a limit of a constant inverse spectrum ${\bf S}_{2}=\{{\cal C}_{1},id,\{1\}\}$ with ${\cal C}_{1}={\cal C}$ and $\Upsilon=\{1\}$, where $id$ denotes the identity homomorphism, where the microbundle ${\cal C}$ is given for some left module $Y$ over a topological ring $\bf G$. Therefore a family $T=\{t_{k}:~{}k\in\Lambda\}$ is a homomorphism from ${\cal C}$ into $\bf S$.

Let $(a,e,f,z)\in Q:=\{(b,d,q,x)\in A\times E\times{\bf F}\times X:~{}p(d)=b\}$. We take a neighborhood $R:=\{U\times V\times P\times S:~{}p(V)=U\}$ of $(a,e,f,z)$, where $U$ is an open subset in a base space $A$, $~{}V$ is an open subset in a total space $E$, $~{}P$ is an open subset in a topological ring $\bf F$, $~{}S$ is an open subset in a topological left $\bf F$ module $X$. Since $t_{k}(C,D,{\bf G},Y)$ is dense in $Q_{k}:=\{(a_{k},e_{k},f_{k},z_{k})\in A_{k}\times E_{k}\times{\bf F}_{k}\times X_{k}:~{}p_{k}(e_{k})=a_{k}\}$, then an open subset $R_{k}:=\{U_{k}\times V_{k}\times P_{k}\times S_{k}:~{}p_{k}(V_{k})=U_{k}\}$ in $Q_{k}$ exists such that $(a,e,f,z)\in\pi_{k}^{-1}(R_{k})$ and $\pi_{k}^{-1}(R_{k})\subset R$. There exists $(c,d,g,y)\in W:=\{(b,d,q,v)\in C\times D\times{\bf G}\times Y:~{}\breve{p}(d)=b\}$ such that $t_{k}(c,d,g,y)\in R_{k}$. Hence $t(c,d,g,y)\in\pi_{k}^{-1}(R_{k})$ and consequently, $t({\cal C})$ is dense in ${\cal B}(A,E,{\bf F},X,i,p)$.

31. Proposition. Let $(id,T):{\bf S}_{1}\to{\bf S}_{2}$ be a homomorphism of inverse spectra ${\bf S}_{j}=\{{\cal B}_{j,k},~{}\mbox{}^{j}\pi^{n}_{k}:k\in\Lambda\}$ with $j\in\{1,2\}$ of microbundles ${\cal B}_{j,k}$ and let $t_{k}:{\cal B}_{1,k}\hookrightarrow{\cal B}_{2,k}$ be an embedding for each $k\in\Lambda$. Then $\mbox{}^{1}\pi^{n}_{k}=\mbox{}^{2}\pi^{n}_{k}\circ t_{k}|_{{\cal B}_{1,k}}$ for each $k\leq n$ in $\Lambda$ and $\lim{\bf S}_{1}=\bigcap_{k\in\Lambda}\mbox{}^{2}\pi_{k}^{-1}(t_{k}({\cal B}_{1,k}))$.

Proof. From the condition $\mbox{}^{2}\pi^{n}_{k}\circ t_{n}=t_{k}\circ\mbox{ }^{1}\pi^{n}_{k}$ for each $k\leq n\in\Lambda$ and Theorem 29 it follows that $\mbox{}^{1}\pi^{n}_{k}=\mbox{}^{2}\pi^{n}_{k}\circ t_{n}|_{{\cal B}_{1,n}}$ for each $k\leq n$ in $\Lambda$, since $t_{k}$ is the embedding. Then a limit $t:=\lim T$ is a bijective map from ${\cal B}_{1}=\lim{\bf S}_{1}$ into ${\cal B}_{2}=\lim{\bf S}_{2}$. For an element $(a,b,c,x)\in{\cal B}_{1}$ take a neighborhood $W$. A neighborhood $V_{k}$ of $\mbox{}^{1}\pi_{k}(a,b,c,x)$ in ${\cal B}_{1,k}$ exists such that $(\mbox{}^{1}\pi_{k})^{-1}(V_{k})\subset W$, consequently, $V:=(\mbox{}^{2}\pi_{k})^{-1}(t_{k}(V_{k}))$ is a neighborhood of $t(a,b,c,x)$ in $t({\cal B}_{1})$ such that $t^{-1}(V)\subset W$. Therefore ${\cal B}_{1}\subset{\cal B}_{2}$ and ${\cal B}_{1}\subset\bigcap_{k\in\Lambda}\mbox{}^{2}\pi_{k}^{-1}(t_{k}({\cal B}_{1,k}))$. On the other hand, if $(d,e,q,y)\in\bigcap_{k\in\Lambda}\mbox{}^{2}\pi_{k}^{-1}(d_{k},e_{k},q_{k},x_{k})$, then a family $\{\mbox{}^{2}\pi_{k}(d,e,q,y):~{}k\in\Lambda\}$ is a threat of the spectrum ${\cal S}_{1}$, consequently, $\bigcap_{k\in\Lambda}\mbox{}^{2}\pi_{k}^{-1}(t_{k}({\cal B}_{1,k}))\subset{\cal B}_{1}$.

32. Theorem. Let $s_{1}:X_{\bf F}\to X_{\bf G}$ and $s_{2}:{\bf F}\to{\bf G}$ be homomorphisms of left modules $X_{\bf F}$ and $X_{\bf G}$ and of rings $\bf F$ and $\bf G$ correspondingly such that

$(32.1)$* $s_{1}(v_{1}x_{1}+v_{2}x_{2})=s_{2}(v_{1})s_{1}(x_{1})+s_{2}(v_{2})s_{1}(x_{2})$

for each $v_{1}$ and $v_{2}$ in $\bf F$, $~{}x_{1}$ and $x_{2}$ in $X_{\bf F}$. Let also ${\cal B}(A,E,{\bf F},X_{\bf F},i,p)$ be a microbundle over $\bf F$. Then a microbundle ${\cal B}(A,E^{\prime},{\bf G},X_{\bf G},i^{\prime},p^{\prime})$ exists such that there is a homomorphism $\pi:{\cal B}(A,E,{\bf F},X_{\bf F},i,p)\to{\cal B}(A,E^{\prime},{\bf G},X_{\bf G},i^{\prime},p^{\prime})$ with $s_{1}\circ\hat{\pi}_{2}=\hat{\pi}_{2}^{\prime}\circ\pi^{2}$. Moreover, if $s_{1}$ and $s_{2}$ are either surjective or bijective, then ${\cal B}(A,E^{\prime},{\bf G},X_{\bf G},i^{\prime},p^{\prime})$ and $\pi$ can be chosen such that the homomorphism $\pi$ is either surjective or bijective correspondingly.*

Proof. Let $e\in E$ and $b\in A$ be such that $i(b)=e$. We take a neighborhood $V$ of $e$ homeomorphic with $U\times X_{\bf F}$, where $h$ is a homeomorphism from $V$ onto $U\times X_{\bf F}$, where $U$ is a neighborhood of $b$ in $A$. We put $\pi_{2}(V)=V^{\prime}$ to be homeomorphic with $U\times X_{\bf G}$ with a homeomorphism $h^{\prime}:V^{\prime}\to U\times X_{\bf G}$ and projections $\hat{\pi}_{1}^{\prime}:V^{\prime}\to U$, $~{}\hat{\pi}_{2}^{\prime}:V^{\prime}\to X_{\bf G}$. This implies that $s_{1}\circ\hat{\pi}_{2}|_{V}=\hat{\pi}_{2}^{\prime}\circ\pi^{2}|_{V}$ (see also the notation $(29.3)$ and $(29.4)$). Therefore substituting $V$ on $V^{\prime}$ and $h$ on $h^{\prime}$ and $X_{\bf F}$ on $X_{\bf G}$ induces maps $i^{\prime}|_{U}$, $p^{\prime}|_{V^{\prime}}$, $\iota_{0}^{\prime}|_{U}$ such that $p^{\prime}|_{V^{\prime}}\circ i^{\prime}|_{U}=\hat{\pi}_{1}^{\prime}\circ\iota_{0}^{\prime}|_{U}$ and $\hat{\pi}_{1}^{\prime}\circ h^{\prime}|_{V^{\prime}}=p^{\prime}|_{V^{\prime}}$.

If $U$ and $U_{1}$ are open neighborhoods of $b$ and $b_{1}$ in $A$, $~{}V=h^{-1}(U\times X_{\bf F})$ and $V_{1}=h^{-1}(U_{1}\times X_{\bf F})$ then $\pi^{2}(V)\cap\pi^{2}(V_{1})=\pi^{2}(V\cap V_{1})={h^{\prime}}^{-1}((U\cap U_{1})\times X_{\bf G})$. This provides and equivalence relation $\Xi$ for each $v\in V^{\prime}$ and $v_{1}\in{V_{1}}^{\prime}$: $v\Xi v_{1}$ if and only if $\hat{\pi}_{1}^{\prime}\circ h^{\prime}(v)=\hat{\pi}_{1}^{\prime}\circ h^{\prime}(v_{1})$ and $\hat{\pi}_{2}^{\prime}\circ h^{\prime}(v)=\hat{\pi}_{2}^{\prime}\circ h^{\prime}(v_{1})$. Using the latter property we choose as a total space

$(32.2)$ $E^{\prime}=\bigcup\{V^{\prime}:~{}\exists U,~{}U\mbox{ is open in }A,~{}V^{\prime}=\pi^{2}(V),~{}V=h^{-1}(U\times X_{\bf F})\}/\Xi$.

Bases of topologies on $A$ and $X_{\bf G}$ induce a base of a topology on $E^{\prime}$. Hence a microbundle ${\cal B}(A,E^{\prime},{\bf G},X_{\bf G},i^{\prime},p^{\prime})$ exists.

Then we put $\pi^{1}=id$, $~{}\pi^{3}=s_{2}$, $~{}\pi^{4}=s_{1}$ and take a combination

$\pi^{2}=\nabla\{\pi^{2}|_{V}:~{}\exists U,~{}U\mbox{ is open in }A,~{}V=h^{-1}(U\times X_{\bf F})\}$.

From Condition $(32.1)$ it follows that

$(32.3)$ $\pi^{2}\circ i=i^{\prime}\circ\pi^{1}$ and $p^{\prime}\circ\pi^{2}=\pi^{1}\circ p$ and

$(32.4)$ $h^{\prime}\circ\pi^{2}\circ h^{-1}(b,v_{1}x_{1}+v_{2}x_{2})=$

$\pi^{3}(v_{1})h^{\prime}\circ\pi^{2}\circ h^{-1}(b,x_{1})+\pi^{3}(v_{2})h^{\prime}\circ\pi^{2}\circ h^{-1}(b,x_{2})$ and

$(32.5)$ ${\hat{\pi}^{\prime}}_{2}\circ{h^{\prime}}\circ\pi^{2}=\pi^{4}\circ\hat{\pi}_{2}\circ h$

for every $v_{1}$ and $v_{2}$ in ${\bf F}$, $x_{1}$ and $x_{2}$ in $X_{\bf F}$, $b\in A$. Thus from $(32.3)$ and $(32.4)$ this provides a homomorphism $\pi=(\pi^{1},\pi^{2},\pi^{3},\pi^{4})$ from ${\cal B}(A,E,{\bf F},X_{\bf F},i,p)$ into ${\cal B}(A,E^{\prime},{\bf G},X_{\bf G},i^{\prime},p^{\prime})$.

In particular, if $s_{1}$ and $s_{2}$ are surjective, $s_{1}(X_{\bf F})=X_{\bf G}$ and $s_{2}({\bf F})=\bf G$, then from Formula $(32.2)$ it follows that $\pi^{2}(E)=E^{\prime}$, consequently, $\pi$ is surjective. If $s_{1}$ and $s_{2}$ are bijective, then from Identities $(32.3)$-$(32.5)$ we infer that $\pi^{2}:E\to E^{\prime}$ is bijective and hence $\pi$ is bijective.

33. Corollary. If Conditions of Theorem 32 are satisfied and a microbundle ${\cal B}(A,E^{\prime},{\bf G},X_{\bf G},i^{\prime},p^{\prime})$ is provided by this theorem and $s_{1}$ and $s_{2}$ are isomorphisms, then $\pi$ is an isomorphism of a microbundle ${\cal B}(A,E,{\bf F},X_{\bf F},i,p)$ with ${\cal B}(A,E^{\prime},{\bf G},X_{\bf G},i^{\prime},p^{\prime})$. Particularly, if $s_{1}:X_{\bf F}\to X_{\bf F}$ and $s_{2}:{\bf F}\to\bf F$ are automorphisms, then $\pi$ is an automorphism of ${\cal B}(A,E,{\bf F},X_{\bf F},i,p)$.

34. Theorem. Let $A$ be a Tychonoff base space and let ${\cal B}(A,E,{\bf F},X,i,p)$ be a microbundle. Then for each compactification $cA$ of $A$ there exists a microbundle ${\cal B}(cA,E^{\prime},{\bf F},X,i^{\prime},p^{\prime})$ and an embedding $\pi:{\cal B}(A,E,{\bf F},X,i,p)\hookrightarrow{\cal B}(cA,E^{\prime},{\bf F},X,i^{\prime},p^{\prime})$ such that $i^{\prime}\circ\pi^{1}=\pi^{2}\circ i$, $~{}p^{\prime}\circ\pi^{2}=\pi^{1}\circ p$, $~{}\pi^{3}=I$, $~{}\pi^{4}=I_{X}$, where $I$ denotes the identity map on a ring ${\bf F}$, $~{}I_{X}$ is the identity map on a left ${\bf F}$ module $X=X_{\bf F}$, $~{}\pi=(\pi^{1},\pi^{2},\pi^{3},\pi^{4})$, $~{}\pi^{1}:A\to cA$, $~{}\pi^{2}:E\to E^{\prime}$, $E^{\prime}={E^{\prime}}_{c}$. Moreover, ${\cal B}(\beta A,{E^{\prime}}_{\beta},{\bf F},X,i^{\prime},p^{\prime})$ is a maximal microbundle among such extensions of ${\cal B}(A,E,{\bf F},X,i,p)$, where $\beta A$ denotes the Stone-$\check{C}$ech compactification of $A$.

Proof. By virtue of Theorem 3.5.1 in [3] there exists a compactification $cA$ of $A$. Take an arbitrary fixed point $b\in A$. There exists a neighborhood $U$ of $b$ in $A$ such that $i|_{U}:U\to E$ is continuous with $p\circ i|_{U}=id|_{U}$ and $\iota_{0}|_{U}:U\to U\times X_{\bf F}$ is an embedding. Then $h:V\to U\times X_{\bf F}$ is a homeomorphism, where $V=i(U)$.

Take $U^{\prime}$ open in $cA$ such that $U^{\prime}\cap c(A)=c(U)$, consequently, $(U^{\prime}\times X_{\bf F})\cap(c(A)\times X_{\bf F})=c(U)\times X_{\bf F}$, where $c:A\hookrightarrow cA$ denotes a homeomorphic embedding. Therefore, a topological space $V^{\prime}$ and a homeomorphism $h^{\prime}:V^{\prime}\to U^{\prime}\times X_{\bf F}$ exist such that ${h^{\prime}}^{-1}\circ(c\times I_{X})|_{U\times X_{\bf F}}=h^{-1}|_{U\times X_{\bf F}}$, where $I_{X}$ denotes an identity map on $X_{\bf F}$. Evidently, there are extensions $i^{\prime}:U^{\prime}\to V^{\prime}$ and $p^{\prime}:V^{\prime}\to U^{\prime}$ and $\iota_{0}^{\prime}:U^{\prime}\to U^{\prime}\times X_{\bf F}$ and $\hat{\pi}_{1}^{\prime}:U\times X_{\bf F}\to U^{\prime}$ and $\hat{\pi}_{2}^{\prime}:U^{\prime}\times X_{\bf F}\to X_{\bf F}$ such that $p^{\prime}\circ i^{\prime}|_{U^{\prime}}=id|_{U^{\prime}}$ and $\hat{\pi}_{1}^{\prime}\circ\iota_{0}^{\prime}|_{U^{\prime}}=p^{\prime}|_{V^{\prime}}\circ i^{\prime}|_{U^{\prime}}$, where $\iota_{0}^{\prime}(b)=(b,0)$ for each $b\in U^{\prime}$, $\hat{\pi}_{1}^{\prime}(b,x)=b$ and $\hat{\pi}_{2}^{\prime}(b,x)=x$ for each $(b,x)\in U^{\prime}\times X_{\bf F}$.

Then $V^{\prime}\cap V_{1}^{\prime}={h^{\prime}}^{-1}|_{(U^{\prime}\cap U_{1}^{\prime})\times X_{\bf F}}$, where $U_{1}$ is an open neighborhood of $b_{1}\in A$ such that $i|_{U_{1}}:U_{1}\to E$ is continuous with $p\circ i|_{U_{1}}=id|_{U_{1}}$ and $\iota_{0}|_{U_{1}}:U_{1}\to U_{1}\times X_{\bf F}$ is an embedding. This induces an equivalence relation $v\Xi_{1}v_{1}$ for each $v\in V^{\prime}$ and $v_{1}\in V_{1}^{\prime}$ if and only if $\hat{\pi}_{1}^{\prime}\circ h^{\prime}(v)=\hat{\pi}_{1}^{\prime}\circ h^{\prime}(v_{1})$ and $\hat{\pi}_{2}^{\prime}\circ h^{\prime}(v)=\hat{\pi}_{2}^{\prime}\circ h^{\prime}(v_{1})$. We put

$(34.1)$ $E^{\prime}=\bigcup\{V^{\prime}:~{}\exists U^{\prime},~{}U^{\prime}\mbox{ is open in }cA,~{}V^{\prime}={h^{\prime}}^{-1}(U^{\prime}\times X_{\bf F})\}/\Xi_{1}$.

Therefore there exists an embedding $\pi^{2}:E\to E^{\prime}$ such that $h^{\prime}\circ\pi^{2}(V)=c(U)\times X_{\bf F}$ for each open $U$ and $V$ as described above. Taking $\pi^{1}=c$ and $\pi^{3}=I$ and $\pi^{4}=I_{X}$, we deduce from $(34.1)$ and the construction above that $i^{\prime}\circ\pi^{1}=\pi^{2}\circ i$, $~{}p^{\prime}\circ\pi^{2}=\pi^{1}\circ p$.

In view of Theorem 3.6.1 in [3] there is an embedding of

${\cal B}(cA,{E^{\prime}}_{c},{\bf F},X,i^{\prime},p^{\prime})\hookrightarrow{\cal B}(\beta A,{E^{\prime}}_{\beta},{\bf F},X,i^{\prime},p^{\prime})$

for each compactification $cA$ of $A$, consequently, ${\cal B}(\beta A,{E^{\prime}}_{\beta},{\bf F},X,i^{\prime},p^{\prime})$ is maximal among such extensions of the microbundle ${\cal B}(A,E,{\bf F},X,i,p)$.

35. Corollary. Suppose that the conditions of Theorem 34 are satisfied and a ring $\bf F$ and a left module $X_{\bf F}=X$ have compactifications $c_{3,1}\bf F$ and $c_{3,2}X_{\bf F}$ which are a ring $\bf G$ and a left module isomorphic with a left module $X_{\bf G}$ over $\bf G$ respectively. Then there exists a microbundle ${\cal B}(A^{\prime},E^{\prime},{\bf G},X_{\bf G},i^{\prime},p^{\prime})$ such that $A^{\prime}=c_{1}A$ and $E^{\prime}=c_{2}E$ are compactifications of $A$ and $E$ respectively and an embedding ${\bf c}:{\cal B}(A,E,{\bf F},X_{\bf F},i,p)\hookrightarrow{\cal B}(A^{\prime},E^{\prime},{\bf G},X_{\bf G},i^{\prime},p^{\prime})$ exists.

Proof. A procedure of taking an extension ${\cal B}_{1}$ of the microbundle

${\cal B}(A,E,{\bf F},X_{\bf F},i,p)$ at first by Theorem 32 and then extending ${\cal B}_{1}$ by Theorem 34 provides the microbundle ${\cal B}(A^{\prime},E^{\prime},{\bf G},X_{\bf G},i^{\prime},p^{\prime})$ with a compact base space $A^{\prime}$. From Formulas $(32.2)$ and $(34.1)$ we deduce that a total space $E^{\prime}$ is compact, since $A^{\prime}$, $~{}\bf G$ and $X_{\bf G}$ are compact.

37. Definition. Let ${\cal S}=\{{\cal B}_{j},\pi^{k}_{j},J\}$ be an inverse spectrum of microbundles ${\cal B}_{j}={\cal B}(A_{j},E_{j},i_{j},p_{j})$, and let

$\lim\{\pi^{k}_{j}:~{}j\in J,~{}k\in J,~{}j\leq k<l\}:~{}{\cal B}_{l}\to\lim\{{\cal B}_{j},\pi^{k}_{j},J:j\leq k<l\}$

be a homeomorphism for each limit element $l\in J$, then ${\cal S}$ is called continuous.

Let $\tau$ be an infinite cardinal. A directed set $J$ is called $\tau$-complete, if each its linearly ordered subset $K$ of the cardinality $card(K)\leq\tau$ has a supremum in $J$.

If an inverse spectrum ${\cal S}$ of microbundles ${\cal B}_{j}={\cal B}(A_{j},E_{j},{\bf F}_{j},X_{j},i_{j},p_{j})$ with a $\tau$-complete directed set $J$ is continuous and there is a least element $j_{0}$ in $J$ and $\pi^{k}_{j}({\cal B}_{k})={\cal B}_{j}$ with a compact base space $A_{j}$ for each $j\leq k\in J$, then $\cal S$ is called $\tau$-complete.

Assume that an inverse spectrum ${\cal S}$ of microbundles is $\tau$-complete and $wA_{j}\leq\tau$ for each $j\in J$, where $wA_{j}$ denotes a weight of $A_{j}$, then $\cal S$ is called a $\tau$-spectrum.

38. Proposition. Let ${\cal S}_{1}=\{{\cal B}_{1,k},~{}\mbox{}^{1}\pi^{n}_{k}:k\in\Lambda\}$ be a continuous $\tau$-complete spectrum of microbundles and ${\cal S}_{2}=\{{\cal B}_{2,k},~{}\mbox{}^{2}\pi^{n}_{k}:k\in\Lambda\}$ be a $\tau$-spectrum of microbundles, ${\cal B}_{j}=\lim{\cal S}_{j}$, $f_{j_{0}}:{\cal B}_{1,j_{0}}\to{\cal B}_{2,j_{0}}$ and $f:{\cal B}_{1}\to{\cal B}_{2}$ be homomorphisms of microbundles such that $\pi_{2,j_{0}}\circ f=f_{j_{0}}\circ\pi_{1,j_{0}}$. Let also either $E_{j,k}$ be compact for each $j$ and $k$, or ${\bf F}_{1,k}={\bf F}_{2,k}$ and $X_{1,k}=X_{2,k}$ for each $k$. Then $f$ is a limit of homomorphisms between cofinal subspectra of ${\cal S}_{1}$ and ${\cal S}_{2}$.

Proof. According to the conditions of this proposition $\mbox{}^{j}\pi^{1,n}_{1,k}(A_{j,n})=A_{j,k}$ and $\mbox{}^{j}\pi^{2,n}_{2,k}(E_{j,n})=E_{j,k}$ and $\mbox{}^{j}\pi^{3,n}_{3,k}({\bf F}_{j,n})={\bf F}_{j,k}$ and $\mbox{}^{j}\pi^{4,n}_{4,k}(X_{j,n})=X_{j,k}$ for each $k\leq n$ in $J$ and $j\in\{1,2\}$. Then $f=(f^{1},f^{2},f^{3},f^{4})$, where $f^{1}:A_{1}\to A_{2}$, $f^{2}:E_{1}\to E_{2}$, $f^{3}:{\bf F}_{1}\to{\bf F}_{2}$, $f^{4}:X_{1}\to X_{2}$, where a microbundle ${\cal B}_{j}={\cal B}(A_{j},E_{j},{\bf F}_{j},X_{j},i_{j},p_{j})$ is on a left module $X_{{\bf F}_{j}}=X_{j}$ over a ring ${\bf F}_{j}$ for each $j\in\{1,2\}$. If $E_{j,k}$ is a compact total space, then a ring ${\bf F}_{j,k}$ and a left module $X_{j,k}$ are compact as follows from Theorem 3.1.10 in [3] and Condition $(2.3)$ above. By virtue of Theorem 40 in [15] (see also [5]) and Theorems 29 and 34 and Proposition 31 above there exists a cofinal subset $\Lambda$ in $J$ such that $f=\lim\{t_{k}:~{}k\in\Lambda\}$, where $t_{k}:{\cal B}_{1,k}\to{\cal B}_{2,k}$ is a homomorphism for each $k\in\Lambda$.

39. Corollary. If the conditions of Theorem 38 are satisfied and $f_{0}$ and $f$ are homeomorphisms, then $f$ is a limit of homeomorphisms between cofinal subspectra of ${\cal S}_{1}$ and ${\cal S}_{2}$.