Inverse systems with simplicial bonding maps and cell structures

TL;DR

This paper constructs inverse systems of simplicial complexes with bonding maps that approximate a topologically complete space, establishing a homotopy equivalence and linking to cell structures.

Contribution

It introduces a novel method to represent topologically complete spaces via inverse systems of simplicial complexes with specific bonding maps.

Findings

Limit space is homotopy equivalent to the original space

Provides a new connection between inverse systems and cell structures

Extends understanding of topological space approximations

Abstract

For a topologically complete space $X$ and a family of closed covers $\mathcal A$ of $X$ satisfying a "local refinement condition" and a "completeness condition," we give a construction of an inverse system $\mathbf{ N}_{\mathcal A}$ of simplicial complexes and simplicial bonding maps such that the limit space $N_{\infty} = \varprojlim \mathbf{N}_{\mathcal A}$ is homotopy equivalent to $X$. A connection with cell structures [2],[3] is discussed

Equations

$${\mathrm{star}}_{\alpha }^{n+1}(S)=\{V\in \alpha |\text{mbox}thereexistsW\in {\text{star}}_{\alpha }^n(S)\text{mbox}suchthatV\cap W\ne \varnothing \}.$$

$$N_{\lambda }^{(0)}=\{v=(V_1,...,V_n)\in {\alpha }_1\times \cdots \times {\alpha }_n\mid V_1\cap \dots \cap V_n\ne \varnothing \},$$

$${\mathrm{cov}}_{\lambda }=\{\wedge v\mid v\in N_{\lambda }^{(0)}\}$$

$${\varphi }_v={\varphi }_{{\alpha }_1,V_1}\cdots {\varphi }_{{\alpha }_n,V_n}.$$

$$\wedge a=\wedge v_1\cap \cdots \cap \wedge v_k.$$

$$w\in {\mathrm{cov}}_{\{\alpha \}},w\cap V\ne \varnothing \Rightarrow x\in w.$$

$$\bigcup {\text{star}}_{{\mathrm{cov}}_{\lambda \cup \{\alpha \}}}^n(x)\subset U.$$

$$c=u\cap a,d=v\cap b,u,v\in {\mathrm{cov}}_{\lambda },a,b\in \alpha .$$

$$a\cap V\supset u\cap v\supset c\cap d\ne \varnothing$$

$$U\cap \wedge z_1(\lambda )\supset U\cap \wedge z_1(\mu )\supset \wedge z_0(\mu )\cap \wedge z_1(\mu )\ne \varnothing .$$

$$\{\pi (z)\}={\cap }_{\lambda }(\wedge z(\lambda ))$$

$$\pi ({\pi }_{\lambda }^{-1}(V_{\lambda }))\subset U.$$

$$\pi (w)\in \wedge z(\lambda )\subset \bigcup {\text{star}}_{{\mathrm{cov}}_{\lambda }}(\pi (z))\subset U$$

$$p_{\lambda }^{-1}(\cup {\text{star}}_{{\mathrm{cov}}_{\lambda }}(\wedge v))\subset \wedge v$$

$$C_{\lambda }=\{w\in N_{\lambda }^{(0)}|x\in \wedge w\}$$

$${\pi }^{-1}(x)=\underleftarrow{\lim }{K}_{\lambda }.$$

$$H_{\lambda }(z,t)=t{p}_{\lambda }(\pi (z))+(1-t){\pi }_{\lambda }(z)$$

$$p^{♯}:\underrightarrow{\lim }([N_{\lambda },K],({\pi }_{\lambda }^{\mu }{)}^{\#};\Lambda )\to [X,K]$$

$$p^{♯}:\underrightarrow{\lim }([F_{\lambda },K],({\pi }_{\lambda }^{\mu }{)}^{\#};\Lambda )\to [X,K]$$

$${\pi }^{♯}:\underrightarrow{\lim }([N_{\lambda },K],({\pi }_{\lambda }^{\mu }{)}^{\#},\Lambda )\to [N_{\infty },K],$$

$${\pi }_{\lambda }^{\mu }({\text{star}}_{r_{\mu }}^2(z(\mu )))\subset {\text{star}}_{r_{\lambda }}(z(\lambda )),$$

$$(\ast )\text{mbox}foreach\lambda ,\text{mbox}thereexists\mu \ge \lambda \text{mbox}suchthat{A}_{\lambda ,\mu }:=\{{\pi }_{\lambda }^{\nu }(y(\nu ))|\nu \ge \mu \}\text{mbox}isafiniteset,$$

$${\pi }_{{\lambda }^0}^{{\lambda }^1}(y({\lambda }^1))\text{mbox}and{\pi }_{{\lambda }^0}^{{\lambda }^2}(y({\lambda }^2))\text{mbox}areadjacent,thatis,\wedge {\pi }_{{\lambda }^0}^{{\lambda }^1}(y({\lambda }^1))\cap \wedge {\pi }_{{\lambda }^0}^{{\lambda }^2}(y({\lambda }^2))\ne \varnothing .$$

$$\begin{array}[]{ll}(\ast\ast)~{}~{}\mbox{for each locally finite, open, normal cover}~{}\mathcal{U}~{}\mbox{of}~{}X,~{}\mbox{there exist}~{}\lambda\in\Lambda~{}\mbox{and}~{}U\in\mathcal{U}\\ \quad\quad\quad~{}\mbox{such that}~{}F_{\lambda}\subset U.\end{array}$$

Full text

The second author is supported by JSPS KAKENHI Grant Number 17K05241. The third named author is partially supported by National Science and Engineering Research Council Discovery Grant.

1. Introduction

The notion of inverse limits and its variants provides a useful scheme for studying topological spaces by polyhedra. For every topological space $X$, there associates an inverse system ${\bf X}=(X_{\lambda},\pi^{\mu}_{\lambda}:X_{\mu}\to X_{\lambda};\Lambda)$ of polyhedra $X_{\lambda}$ and continuous maps $\pi^{\mu}_{\lambda}$ that reflects the topology of $X$. Such a system is typically given by a polyhedral resolution in the sense of [8] which yields an HPol-expansion defining the shape type of $X$. In most cases, the bonding maps $\pi_{\lambda}^{\mu}$ are continuous or piecewise linear: if the polyhedra $X_{\lambda}$’s are equipped with specific triangulations and the bonding maps are required to be simplicial with respect to the triangulations, then the inverse limit $\varprojlim\mathbf{X}$ may not be homeomorphic, but closely related, to $X$. This has been made explicit by Dydak in [4, Theorem 2.8]: for each compact metrizable space $X$, there exist an inverse sequence $\mathbb{X}$ of finite simplicial complexes and simplicial bonding maps, and a hereditary shape equivalence $f:\varprojlim\mathbb{X}\to X$. See also [12].

This note studies a similar construction for non-compact spaces. For a topologically complete space $X$ and a family $\mathcal{A}$ of closed locally finite normal coverings of $X$ satisfying a “ local refinement condition” and a “ completeness condition,” we construct an inverse system ${\bf N}_{\mathcal{A}}=(N_{\lambda},\pi^{\mu}_{\lambda}:\Lambda)$ of simplicial complexes and simplicial bonding maps, a proper map $\pi:N_{\infty}:=\varprojlim{\bf N}_{\mathcal{A}}\to X$, and a continuous map $p:X\to N_{\infty}$ such that $\pi\circ p=\mathrm{id}_{X}$ and $p\circ\pi$ is $\pi$-fiberwise homotopic to $\mathrm{id}_{N_{\infty}}$. In particular $N_{\infty}$ and $X$ have the same homotopy type. Also the family $\mathcal{A}$ naturally defines an inverse system ${\bf F}_{\mathcal{A}}=(F_{\lambda},\pi^{\mu}_{\lambda}:\Lambda)$, where each $F_{\lambda}$ is a flag complex which contains $N_{\lambda}$ as its subcomplex and each $\pi_{\lambda}^{\mu}$ is a simplicial map, so that $N_{\infty}=\varprojlim{\bf F}_{\mathcal{A}}$. Since flag complexes are uniquely determined by their 1-skeletons, we could say that the space $N_{\infty}$ is induced by an inverse sequence of graphs and graph homomorphisms.

The above result gives a connection of inverse systems of simplicial bonding maps with cell-structures of topological spaces. Cell structures first appeared in unpublished notes of Dębski in 1990. They are inverse systems of vertex sets of graphs and graph homomorphisms (see [2] and [3]). Every topological space admits a cell structure that can be thought of as a system of discrete approximations of the space. It was proved in [3, Theorem 2] that cell structures determine topologically complete spaces and continuous maps. The above inverse system ${\bf F}_{\mathcal{A}}=(F_{\lambda},\pi^{\mu}_{\lambda}:\Lambda)$ naturally defines a cell-structure. When the space $X$ is compact Hausdorff, then the system of projections ${\bf p}=(p_{\lambda}:N_{\infty}\to F_{\lambda})$ forms an HPol-expansion of the space $N_{\infty}$ and thus ${\bf p}\circ p=(p_{\lambda}\circ p)$ is an HPol-expansion of $X$. Thus we could say that an HPol-expansion of a compact Hausdorff space is obtained from a cell-structure.

Our construction is motivated by a theorem of Mardešić [6] (see also [8, Chap.I, Section 6.4, Theorem 7]) stating that every topological space admits a polyhedral resolution. For each topologically complete space $X$ with a family $\mathcal{A}$ of closed locally finite normal covers of $X$ satisfying the above-mentioned conditions, we modify the construction to obtain inverse systems ${\bf N}_{\mathcal{A}}=(N_{\lambda},\pi^{\mu}_{\lambda}:N_{\mu}\to N_{\lambda})$ and $\bf{F}_{\mathcal{A}}$ with the desired properties.

Throughout, for a subset $S$ of a topological space $Y$, $\operatorname{int}(S)$ and $\operatorname{cl}(S)$ denote the interior and the closure of $S$ in $Y$. For a cover $\alpha$ of $Y$, let $\text{star}_{\alpha}(S)=\{V\in\alpha~{}|~{}S\cap V\neq\emptyset\}$ and define inductively

[TABLE]

Simplicial complexes are assumed to be endowed with specific triangulations and the weak topology with respect to the triangulations. For a simplicial complex $N$, $N^{(i)}$ denotes the $i$-skeleton of $N$. In particular $N^{(0)}$ is the set of the vertices of $N$. For a simplex $\sigma$ of $N$, $\text{Int}(\sigma)$ denotes the interior of $\sigma$, the points of $\sigma$ whose barycentric coordinates are all positive. Following [1] we say that a simplicial complex $F$ is a flag complex if every subset $\{v_{1},\ldots,v_{n}\}$ of $F^{(0)}$ with the property: $v_{i},v_{j}$ span a 1-simplex for each $i\neq j$, spans a simplex of $F$. The first barycentric subdivision of each simplicial complex is a flag complex. The reader may consult [5] for standard results in general topology and [8] for results in shape theory.

2. Construction of polyhedral inverse systems with simplicial bonding maps

Recall that a topologically complete space is a Tychonoff space which is complete in its finest uniformity. A closed locally finite cover $\alpha$ is called normal if there exists a partition of unity $\Phi_{\alpha}=\{\phi_{\alpha,V}\mid V\in\alpha\}$ subordinated to $\alpha$: each $\phi_{\alpha,V}:X\to[0,1]$ is continuous, satisfies $\mathrm{supp}(\phi_{\alpha,V})=\mathrm{cl}(\phi_{\alpha,V}^{-1}((0,1]))\subset\mathrm{int}(V)$, and $\sum_{V\in\alpha}\phi_{\alpha,V}=1$. In the remainder of this section we fix an arbitrary topologically complete space $X$ and fix a family $\mathcal{A}$ of closed, locally finite, normal covers of $X$ satisfying the following two conditions:

1. I)

For each open subset $U$ of $X$ and for each $x\in U$, there exists $\alpha\in A$ such that $\bigcup\mathrm{star}_{\alpha}(x)\subset U$. 2. II)

For each $\alpha\in\mathcal{A}$ select a member $f(\alpha)\in\alpha$. If the collection $\{f(\alpha)\}$ has the finite intersection property, then $\bigcap_{\alpha\in A}f(\alpha)\neq\emptyset.$

It should be mentioned here that the family $\mathcal{A}$ need not be cofinal with respect to the refinement-order: a closed locally finite normal covering $\gamma$ of $X$ may not admit $\alpha\in{\mathcal{A}}$ that refines $\gamma$. the condition I), a “local refinement” requirement, is a substitute for the cofinality.

Proposition 2.1**.**

If $Y$ is a closed subset of $X$ then the conditions I) and II) are satisfied for the family $\mathcal{A}|Y=\{\alpha|Y~{}|~{}\alpha\in\mathcal{A}\}$, where $\alpha|Y=\{F\cap Y~{}|~{}F\in\alpha\}$.

Proof.

The condition I) is clearly satisfied. Let $f:A\rightarrow\bigcup A$ be a selection such that the sets $f(\alpha)\cap Y$ have the finite intersection property. Then by II) $\bigcap_{\alpha\in A}f(\alpha)\neq\emptyset$. Let $x\in\bigcap_{\alpha\in A}f(\alpha)$. Assume $x\notin Y$. Then by I) there exists $\alpha\in A$ such that $f(\alpha)\subset X\setminus Y$. Thus, $f(\alpha)\cap Y=\emptyset$. This is a contradiction. ∎

2.1. Construction

Here we construct an inverse system $(F_{\lambda},\pi_{\lambda}^{\mu};\Lambda)$ with the limit space $F_{\infty}$ and a proper map $\pi:F_{\infty}\to X$ that is a homotopy equivalence.

Let $\Lambda$ be the directed set of all finite subsets of $\mathcal{A}$ ordered by inclusion. It is a cofinite set in that each element of $\Lambda$ has only finitely many predecessors. For an element $\lambda=\{\alpha_{1},\ldots,\alpha_{n}\}$ of $\Lambda$, let

[TABLE]

and for $v=(V_{1},\cdots,V_{n})\in N^{(0)}_{\lambda}$, let $\wedge v=V_{1}\cap\ldots\cap V_{n}$. Define the collection $\operatorname{cov}_{\lambda}$ by

[TABLE]

which is a closed, locally finite, normal cover of $X$ indexed by the set $N_{\lambda}^{(0)}$; the local finiteness follows from that of $\alpha_{1},\ldots,\alpha_{n}$. For the existence of a partition of unity subordinated to $\operatorname{cov}_{\lambda}$, let $\{\phi_{\alpha_{i},V}~{}|~{}V\in\alpha_{i}\}$ be a partition of unity subordinated to $\alpha_{i}$. For each $v\in N_{\lambda}^{(0)}$ with $v=(V_{1},\ldots,V_{n})$, let

[TABLE]

It is readily verified that $\{\phi_{v}~{}|~{}v\in N^{(0)}_{\lambda}$} is a partition of unity subordinated to the cover $\operatorname{cov}_{\lambda}$ indexed by $N^{(0)}_{\lambda}$. Note that it may be the case $\wedge v=\wedge w$ for distinct elements $v$ and $w$ of $N_{\lambda}^{(0)}$.

Let $F_{\lambda}$ be the flag complex defined as follows: the vertex set of $F_{\lambda}$ is the set $N_{\lambda}^{(0)}$; and a finite set $\{v_{1},v_{2},...,v_{n}\}$, $v_{i}\in N^{(0)}_{\lambda}$, spans a simplex of $F_{\lambda}$ if and only if $\wedge v_{i}\cap\wedge v_{j}\neq\emptyset$ for each pair $i,j\in\{1,...,n\}$. For each point $a$ of $F_{\lambda}$, there exists a unique simplex $\sigma_{\lambda}(a)$ of $F_{\lambda}$ such that $a\in\text{Int}\sigma_{\lambda}(a)$. If $\sigma_{\lambda}(a)$ has the vertex set $\{v_{1},\ldots,v_{n}\}$ of $F_{\lambda}^{(0)}$, then let $\wedge a$ be the subset of $X$ (maybe empty) defined by

[TABLE]

Let $N_{\lambda}$ be the subcomplex of $F_{\lambda}$ defined as follows: the set of vertices is equal to $N_{\lambda}^{(0)}$ and a finite set $\{v_{1},v_{2},...,v_{n}\}$ of vertices spans a simplex in $N_{\lambda}$ if and only if $\bigcap_{i=1}^{n}\wedge v_{i}\neq\emptyset$. By definition, we have $N_{\lambda}^{(1)}=F_{\lambda}^{(1)}$. Also for a point $a\in F_{\lambda}$, $\wedge a\not=\emptyset$ if and only if $a\in N_{\lambda}$. The complex $N_{\lambda}$ is almost the same as the nerve complex $\mathcal{N}(\operatorname{cov}_{\lambda})$ of $\operatorname{cov}_{\lambda}$, except that distinct vertices of $N_{\lambda}$ may define the same vertex of $\mathcal{N}(\operatorname{cov}_{\lambda})$. There exists a natural simplicial homotopy equivalence $N_{\lambda}\to\mathcal{N}(\operatorname{cov}_{\lambda})$ such that the inverse image of each simplex of $\mathcal{N}(\operatorname{cov}_{\lambda})$ is a simplex.

For two elements $\lambda\leq\mu$ of $\Lambda$, written as $\lambda=\{\alpha_{1},\ldots,\alpha_{n}\}$ and $\mu=\{\alpha_{1},\ldots,\alpha_{n},\ldots,\alpha_{m}\}$, let $\pi^{\mu}_{\lambda}:F_{\mu}\rightarrow F_{\lambda}$ be the simplicial map which sends each vertex $(V_{1},\ldots,V_{n},\ldots,V_{m})\in\alpha_{1}\times\cdots\times\alpha_{m}$ of $F^{(0)}_{\mu}$ to the vertex $(V_{1},\ldots,V_{n})$ of $F^{(0)}_{\lambda}$. We see $\pi^{\nu}_{\lambda}=\pi^{\mu}_{\lambda}\circ\pi^{\nu}_{\mu},~{}~{}\lambda\leq\mu\leq\nu,$ and obtain an inverse system ${\bf F}_{\mathcal{A}}=(F_{\lambda},\pi_{\lambda}^{\mu};\Lambda)$. Due to the inclusion $\pi^{\mu}_{\lambda}(N_{\mu})\subset N_{\lambda}$, we have a subsystem ${\bf N}_{\mathcal{A}}=(N_{\lambda},\pi_{\lambda}^{\mu};\Lambda)$. Let $F_{\infty}=\varprojlim\bf{F}_{\mathcal{A}}$, $N_{\infty}=\varprojlim\bf{N}_{\mathcal{A}}$. For $\lambda\in\Lambda$, let $\pi_{\lambda}:F_{\infty}\rightarrow F_{\lambda},\pi_{\lambda}:N_{\infty}\rightarrow N_{\lambda},$ be the $\lambda^{\text{th}}$ coordinate projections. For a point $z\in F_{\infty}$, the point $\pi_{\lambda}(z)$ is also denoted by $z(\lambda)$.

Lemma 2.2**.**

Under the above notation, we have the following.

• (1)

For each open subset $U$ of $X$, for each $x\in U$ and for each positive integer $n$, there exists $\lambda\in\Lambda$ such that $\bigcup\emph{star}^{n}_{\operatorname{cov}_{\lambda}}(x)\subset U$.

• (2)

For each $z\in F_{\infty}$, the set $\bigcap_{\lambda\in\Lambda}(\wedge z(\lambda))$ contains exactly one point.

Proof.

(1) We use induction on $n$. The case $n=1$ follows from the condition I). Assume that (1) holds for $(n-1)$ and apply the condition I) to take $\alpha\in{\mathcal{A}}$ such that $\bigcup\text{star}_{\operatorname{cov}_{\alpha}}(x)\subset U$. By the local finiteness of the covering $\operatorname{cov}_{\{\alpha\}}$ there exists an open neighborhood $V$ of $x$ such that

[TABLE]

Use the induction hypothesis to find a $\lambda\in\Lambda$ such that $\bigcup\text{star}^{n-1}_{\lambda}(x)\subset V$. We show

[TABLE]

For each $c\in\text{star}^{n}_{\operatorname{cov}_{\lambda\cup\{\alpha\}}}(x)$, there exists $d\in\text{star}^{n-1}_{\operatorname{cov}_{\lambda\cup\{\alpha\}}}(x)$ such that $c\cap d\neq\emptyset$. The elements $c$ and $d$ are written as

[TABLE]

Then $v$ is in $\text{star}^{n-1}_{\lambda}(x)$ and hence $v\subset V$. Thus the choice of $V$ and the inclusion

[TABLE]

imply that $x\in a$. Hence $a\subset\bigcup\text{star}_{\operatorname{cov}_{\lambda}}(x)\subset U$. Thus $c=u\cap a\subset U$.

(2) Let $z$ be a point of $F_{\infty}$. For $\lambda\in\Lambda$, the point $z(\lambda)$ is an interior point of the unique simplex $\sigma_{\lambda}(z(\lambda))$ of $F_{\lambda}$. Let $\{v_{j}~{}|~{}j=1,\ldots,k\}$ be the set of vertices of $\sigma_{\lambda}(z(\lambda))$ and recall $\wedge z(\lambda)=\cap_{j=1}^{k}\wedge v_{j}$. For each $\lambda,\mu\in\Lambda$ with $\lambda\leq\mu$ we have $\pi^{\mu}_{\lambda}(\sigma_{\mu}(z(\mu))^{(0)})=\sigma_{\lambda}(z(\lambda))^{(0)}$, and each $\sigma_{\lambda}(z({\lambda}))^{(0)}$ is a finite set. We thus have an inverse system $(\sigma_{\lambda}(z(\lambda))^{(0)},\pi_{\lambda}^{\mu};\Lambda)$ of finite sets and surjective bonding maps so that the limit space $\varprojlim\sigma_{\lambda}(z(\lambda))^{(0)}$ is nonempty and projects onto each $\sigma_{\lambda}(z(\lambda))^{(0)}$. Hence there exists $z_{0}\in F_{\infty}$ such that $z_{0}(\lambda)\ \in\sigma_{\lambda}^{(0)}\subset F_{\lambda}^{(0)}$ for each $\lambda\in\Lambda$. Note that for $\lambda<\mu$, $\wedge z_{0}(\mu)\subset\wedge z_{0}(\lambda)$. Let $\mathcal{F}_{0}=\{\wedge z_{0}(\lambda)~{}|~{}\lambda\in\Lambda\}$. By the above remark, it has the finite intersection property and thus by the condition II), the intersection $\bigcap_{\lambda\in\Lambda}\wedge z_{0}(\lambda)$ is non-empty. If two distinct points $p,q$ are in the set, we take an open neighborhood $U$ of $p$ which does not contain $q$ and apply the condition I) to find $\lambda\in\Lambda$ such that $\bigcup\text{star}_{\operatorname{cov}_{\lambda}}(p)\subset U$. Since $p\in z_{0}(\lambda)$, we see $z_{0}(\lambda)\in\text{star}_{\operatorname{cov}_{\lambda}}(p)$ and thus $\wedge z_{0}(\lambda)\subset U$. But then $q\in\wedge z_{0}(\lambda)\subset U$, a contradiction. This implies that $\bigcap_{\lambda\in\Lambda}\wedge z_{0}(\lambda)$ is a singleton. Let $\{x\}=\bigcap_{\lambda\in\Lambda}\wedge z_{0}(\lambda)$.

Next we show that the above $x$ does not depend on the choice of $z_{0}$. Let $z_{1}$ be another point of $F_{\infty}$ such that $z_{1}(\lambda)\in\sigma_{\lambda}(z)^{(0)}$ for each $\lambda$. Since $z_{0}(\lambda)$ and $z_{1}(\lambda)$ are vertices of $\sigma_{\lambda}(z(\lambda))\in F_{\lambda}$, we have $\wedge z_{0}(\lambda)\cap\wedge z_{1}(\lambda)\neq\emptyset$. Let $U$ be an arbitrary open neighborhood of $x$ and use the condition I) to find $\alpha\in\mathcal{A}$ such that $\bigcup\mathrm{star}_{\alpha}(x)\subset U$. Let $\mu=\lambda\cup\{\alpha\}$ for which we have $\bigcup\mathrm{star}_{\operatorname{cov}_{\mu}}(x)\subset U$. Since $x\in\wedge z_{0}(\mu)$ we have $\wedge z_{0}(\lambda)\in\text{star}_{\operatorname{cov}_{\mu}}(x)$ and hence $\wedge z_{0}(\lambda)\subset U$. We see

[TABLE]

Thus $x\in\operatorname{cl}(\wedge z_{1}(\mu))=\wedge z_{1}(\mu)$. Hence $\{x\}=\cap_{\lambda}(\wedge z_{1}(\lambda))$.

Finally we show $\{x\}=\cap_{\lambda}(\wedge z(\lambda))$. For each vertex $\wedge v$ of $\sigma_{\lambda}(z(\lambda))$, we choose $z_{0}\in F_{\infty}^{(0)}$ such that $z_{0}(\lambda)=\wedge v$ and $z_{0}(\mu)\in\sigma_{\mu}(z_{0}(\mu))^{(0)}$ for each $\mu$. We then have $x\in\wedge v$ by the above. Thus $x\in\wedge z(\lambda)$ for each $\lambda\in\Lambda$, which completes the proof of (2). ∎

The space $N_{\infty}=\varprojlim\bf{N}_{\mathcal{A}}$ is a closed subspace of $F_{\infty}$, but more holds.

Proposition 2.3**.**

We have the equality $N_{\infty}=F_{\infty}$.

Proof.

Let $z=(z(\lambda))_{\lambda}\in F_{\infty}$. By Lemma 2.2 (2) we have $\bigcap_{\lambda}(\wedge z(\lambda))$ is a singleton $\{x\}$. Then for $\lambda\in\Lambda$, each vertex of $\sigma_{\lambda}(z(\lambda))$ contains the point $x$ and thus $\sigma_{\lambda}(z(\lambda))$ is a simplex of the complex $N_{\lambda}$. Hence $z(\lambda)\in N_{\lambda}$ and $z\in N_{\infty}$. ∎

Since $N_{\infty}=F_{\infty}$ is the limit of flag complexes $F_{\lambda}$ which are uniquely determined by their 1-skeletons, we may say that the limit space $N_{\infty}$ is determined by an inverse system of graphs. Notice that the barycentric subdivision $\mathrm{sd}N_{\lambda}$ is always a flag complex and we thus obtain an inverse system $(\mathrm{sd}N_{\lambda},\mathrm{sd}\pi_{\lambda}^{\mu};\Lambda)$ whose limit is homeomorphic to $N_{\infty}$, yet the graph $(\mathrm{sd}N_{\lambda})^{(1)}$ is not isomorphic to $N_{\lambda}^{(1)}$. Here we obtain the flag complex $F_{\lambda}$ with $F_{\lambda}^{(1)}=N_{\lambda}^{(1)}$.

In view of Lemma 2.2 and Proposition 2.3, we define a map $\pi:N_{\infty}\to X$ by

[TABLE]

for $z=(z(\lambda))_{\lambda}\in N_{\infty}=F_{\infty}$. It is convenient to introduce the space $N_{\infty}^{(0)}=\varprojlim(N_{\lambda}^{(0)},\pi_{\lambda}^{\mu}|N_{\mu}^{(0)};\Lambda)$.

Proposition 2.4**.**

1. (1)

The map $\pi$ is continuous. 2. (2)

For points $z,z^{\prime}\in N_{\infty}^{(0)}$, $\pi(z^{\prime})=\pi(z)$ if and only if $\wedge z^{\prime}(\lambda)\in\emph{star}_{\operatorname{cov}_{\lambda}}(\wedge z(\lambda))$ for each $\lambda\in\Lambda$. 3. (3)

For points $z\in N_{\infty}$ and $z_{0}\in N_{\infty}^{(0)}$ such that $z_{0}(\lambda)\in\sigma_{\lambda}(z(\lambda))^{(0)}$ for each $\lambda$, we have $\pi(z)=\pi(z_{0})$.

Proof.

(1) Take a point $z\in N_{\infty}$ and let $U$ be an open neighborhood of $\pi(z)$. Take $\lambda$ such that $\bigcup\text{star}_{\operatorname{cov}_{\lambda}}(\pi(z))\subset U$ by Lemma 2.2 (1). Let $V_{\lambda}=\cup\{\text{Int}\sigma~{}|~{}z(\lambda)\in\sigma\in N_{\lambda}\}$ be the open star of $z(\lambda)$ in $N_{\lambda}$. It is an open neighborhood of $z(\lambda)$ and we show that

[TABLE]

Let $w\in\pi_{\lambda}^{-1}(V_{\lambda})$. For each $\lambda$, we see $\sigma_{\lambda}(w(\lambda))$ contains $z(\lambda)$. Thus $\sigma_{\lambda}(z(\lambda))$ is a face of $\sigma_{\lambda}(w(\lambda))$. This implies $\wedge w(\lambda)\subset\wedge z(\lambda)$ and thus $\pi(w)\in\wedge w(\lambda)\subset\wedge z(\lambda)$. Hence

[TABLE]

and the conclusion follows.

(2) Suppose $\pi(z)=\pi(z^{\prime})$ for $z,z^{\prime}\in N^{(0)}_{\infty}$ and let $x=\pi(z)$. For each $\lambda$ we see $x\in\wedge z(\lambda)\cap\wedge z^{\prime}(\lambda)$. Thus $\wedge z(\lambda)\cap\wedge z^{\prime}(\lambda)\not=\emptyset$ and $\wedge z^{\prime}(\lambda)\in\text{star}_{\operatorname{cov}_{\lambda}}(\wedge z(\lambda))$ for each $\lambda\in\Lambda$.

Conversely let $\wedge z^{\prime}(\lambda)\in\text{star}_{\operatorname{cov}_{\lambda}}(\wedge z(\lambda))$ for each $\lambda\in\Lambda$. Let $x=\pi(z)$, $x^{\prime}=\pi(z^{\prime})$ and assume $x^{\prime}\not=x$. Take disjoint open sets $U$ and $U^{\prime}$ containing $x$ and $x^{\prime}$ respectively. By I) there are $\alpha,\alpha^{\prime}\in\mathcal{A}$ such that $\bigcup\mathrm{star}_{\alpha}(x)\subset U$ and $\bigcup\mathrm{star}_{\alpha}(x^{\prime})\subset U^{\prime}$. Since $x\in\wedge z(\{\alpha\})\subset U$ and $x^{\prime}\in\wedge z(\{\alpha^{\prime}\})\subset U^{\prime}$, we have $\wedge z(\{\alpha\})\cap\wedge z^{\prime}(\{\alpha^{\prime}\})=\emptyset$. Thus $\wedge z(\lambda)\cap\wedge z^{\prime}(\lambda)=\emptyset$ for $\lambda=\{\alpha,\alpha^{\prime}\}$, a contradiction.

(3) Let $z\in N_{\infty}$ and $z_{0}\in N^{(0)}_{\infty}$ such that $z_{0}(\lambda)$ is a vertex of the simplex $\sigma_{\lambda}(z(\lambda))$ for each $\lambda\in\Lambda$. Since $\wedge z({\lambda})\subset\wedge z_{0}({\lambda})$ we have $\pi(z)\in\bigcap_{\lambda\in\Lambda}(\wedge z_{0}(\lambda))$. Thus, $\pi(z)=\pi(z_{0})$. ∎

To study the map $\pi$ further, let us introduce a map $p:X\to F_{\infty}=N_{\infty}$ as follows: for each $\lambda=\{\alpha_{1},\ldots,\alpha_{n}\}\in\Lambda$, we define $p_{\lambda}:X\to N_{\lambda}$ as the canonical map associated with the partition of unity (2.1). It has the property

[TABLE]

for each $\wedge v\in\operatorname{cov}_{\lambda}$. As is proved in [8, p. 85], we have $p_{\lambda}=\pi^{\mu}_{\lambda}\circ p_{\mu}$ for $\lambda\leq\mu$ and hence obtain a system of maps ${\bf p}=(p_{\lambda}):X\to{\bf N}_{\mathcal{A}}=(N_{\lambda},\pi_{\lambda}^{\mu};\Lambda)$ which induces the limit map $p=\varprojlim\mathbf{p}:X\to N_{\infty}$. The map $p$ is continuous and satisfies $p_{\lambda}=\pi_{\lambda}\circ p$ for each $\lambda$.

Theorem 2.5**.**

Under the above notation, we have the following.

• (1)

The map $\pi:N_{\infty}=F_{\infty}\to X$ is a perfect map.

• (2)

We have

1. (2a)

$\pi\circ p=\operatorname{id}_{X}$, and 2. (2b)

there exists a homotopy $H:N_{\infty}\times[0,1]\to N_{\infty}$ such that $H_{0}=\operatorname{id}_{N_{\infty}},H_{1}=p\circ\pi$ and $\pi\circ H=\pi\circ\mathrm{proj}$, where $\mathrm{proj}:N_{\infty}\times[0,1]\to N_{\infty}$ denotes the projection onto $N_{\infty}$. In particular each fiber of $\pi$ is contractible.

Proof.

(1) Fix an arbitrary point $x\in X$. We first prove that $\pi^{-1}(x)$ is non-empty and compact. For each $\lambda=\{\alpha_{1},\ldots,\alpha_{n}\}$, let

[TABLE]

which is a finite subset of $N_{\lambda}^{(0)}$ due to the local finiteness of the covers $\alpha_{1},\ldots\alpha_{n}$. If $\mu\geq\lambda$ then $x\in\wedge\pi_{\lambda}^{\mu}(u)$ and $\pi_{\lambda}^{\mu}(u)\in C_{\lambda}$ for each $u\in C_{\mu}$. Thus $(C_{\lambda},\pi_{\lambda}^{\mu}|C_{\mu};\Lambda)$ forms an inverse system of finite sets and hence has the nonempty inverse limit. If $z=(z(\lambda))\in\varprojlim C_{\lambda}$, then $x\in\wedge z(\lambda)$ and thus $\sigma_{\lambda}(z(\lambda))^{(0)}\subset C_{\lambda}$ for each $\lambda$. By the definition of the map $\pi$, we have $x=\pi(z)$. Next let $K_{\lambda}$ be the simplex of $N_{\lambda}$ spanned by $C_{\lambda}$. We prove

[TABLE]

If $\pi(z)=x$, then $x\in z(\lambda)$ for each $\lambda$, which implies that $\sigma_{\lambda}(z(\lambda))^{(0)}\subset C_{\lambda}$ and hence $z(\lambda)\in K_{\lambda}$. Thus we have $z\in\varprojlim K_{\lambda}$ and $\pi^{-1}(x)\subset\varprojlim K_{\lambda}$. The reverse inclusion is straightforward. This proves the above and hence $\pi^{-1}(x)$ is compact.

Next we show that $\pi$ is a closed map. Let $G$ be a closed subset of $N_{\infty}$ and take an arbitrary point $x\in\operatorname{cl}(\pi(G))$. Under the above notation, we consider the simplex $K_{\lambda}$, $\lambda\in\Lambda$, for the point $x$. First observe that $G_{\lambda}:=K_{\lambda}\cap\operatorname{cl}(\pi_{\lambda}(G))$ is compact as a closed subset of the simplex $K_{\lambda}$. Let $U=X\setminus\bigcup\{\wedge v\mid x\not\in\wedge v,v\in N^{(0)}_{\lambda}\}$. It is an open set containing $x$. It follows that there exists $z\in G$ such that $\pi(z)\in U$. This implies $\wedge z(\lambda)\cap U\neq\emptyset$. For each vertex $v\in\sigma_{\lambda}(z(\lambda))$, we have $x\in\wedge v$ by the definition of $U$ and thus $z(\lambda)\in K_{\lambda}\cap\pi_{\lambda}(G)\subset G_{\lambda}$ and $G_{\lambda}\not=\emptyset$. It follows that $\pi_{\lambda}^{\mu}(G_{\mu})\subset G_{\lambda}$ and hence $\varprojlim(G_{\lambda},\pi_{\lambda}^{\mu})$ is a nonempty compact subset of $N_{\infty}$. Let $z^{\prime}\in\varprojlim(G_{\lambda},\pi^{\mu}_{\lambda})$. Then we have $z^{\prime}\in G$: if not, there exists an open neighborhood $O$ of $z^{\prime}$ such that $G\cap O=\emptyset$ because $G$ is closed. Take an index $\lambda$ and an open neighborhood $V$ of $z(\lambda)$ such that $\pi_{\lambda}^{-1}(V)\subset O$. Then we see that $z^{\prime}(\lambda)\notin\operatorname{cl}\pi_{\lambda}(G)$, a contradiction. Also $x\in\wedge z^{\prime}(\lambda)$ for each $\lambda\in\Lambda$. Thus we see $x=\pi(z^{\prime})$ and hence $\pi(G)$ is closed. This proves (1).

(2) For each $x\in X$ and for each $\lambda$, take any vertex $v\in\sigma_{\lambda}(p_{\lambda}(x))$. We see $p_{\lambda}(x)\in\text{star}_{\operatorname{cov}_{\lambda}}(\wedge v)$ and by (2.2) $x\in p_{\lambda}^{-1}(\text{star}_{\operatorname{cov}_{\lambda}}(\wedge v))\subset\wedge v$. Hence we have $x\in\wedge p_{\lambda}(x)$ for each $\lambda$. It follows from the definition of $\pi$ that $\pi(p(x))=x$. This proves the statement (2.a). To prove the statement (2.b), we define a homotopy $H:N_{\infty}\times I\rightarrow N_{\infty}$ as follows: for a point $z\in N_{\infty}$ and $\lambda$, both $z(\lambda)=\pi_{\lambda}(z)$ and $p_{\lambda}(\pi(z))$ are points of the simplex $\sigma_{\lambda}(z(\lambda))$. Thus

[TABLE]

is a well-defined point of $\sigma_{\lambda}(z(\lambda))$. Also we have $H_{\lambda}(z,t)=\pi_{\lambda}^{\mu}(H_{\mu}(z,t))$ for $\lambda\leq\mu$ because $\pi_{\lambda}^{\mu}$ is a simplicial map. Hence $H(z,t)=(H_{\lambda}(z,t))$ is a well-defined point of $N_{\infty}$. This defines a continuous homotopy between $\operatorname{id}_{N_{\infty}}$ and $p\circ\pi$. In order to verify $\pi\circ H=\pi$, we take $z\in N_{\infty}$ and set $x=\pi(z)$. By the definition of $\pi$, we see $x\in\wedge z(\lambda)$. Let $C_{\lambda}$ be the finite subset defined in (2.3) for $x$ and let $K_{\lambda}$ be the simplex spanned by $C_{\lambda}$. Each vertex of $\sigma_{\lambda}(z(\lambda))$ belongs to $C_{\lambda}$, from which it follows that $\sigma_{\lambda}(z(\lambda))\subset K_{\lambda}$. The equality (2.4) shows that $z\in\varprojlim\sigma_{\lambda}(z(\lambda))\subset\pi^{-1}(x)$. Hence we have $\pi\circ H(z,t)=\pi(z)$. This proves (2.b). ∎

We have a system of inclusions $\mathbf{i}=(i_{\lambda}:N_{\lambda}\hookrightarrow F_{\lambda})$. Proposition 2.3 does not guarantee that $\mathbf{i}$ is an isomorphism in pro-Top (see [8]). When the space $X$ is paracompact and $\mathcal{A}$ is a cofinal family, it is indeed the case.

Theorem 2.6**.**

Assume that $X$ is a paracompact space and $\mathcal{A}$ be a cofinal subfamily of all closed, locally finite, normal covers of $X$. Then for each $\lambda\in\Lambda$ there exists $\mu\in\Lambda$ with $\mu\geq\lambda$ such that $\pi_{\lambda}^{\mu}(F_{\mu})\subset N_{\lambda}$. In particular the system of the inclusions $(N_{\lambda}\hookrightarrow F_{\lambda};\lambda\in\Lambda)$ is an isomorphism in the category pro-Top.

Proof.

Let $\lambda=\{\alpha_{1},\ldots,\alpha_{n}\}$. For each point $x$ of $X$, let $U_{x}=X\setminus\bigcup\{F\in\operatorname{cov}_{\lambda}\mid x\not\in F\}$. Take a locally finite open star refinement $\mathcal{U}$ of $X$ of $\{U_{x}\mid x\in X\}$ and its closed normal locally finite refinement $\mathcal{V}$. There is a cover $\alpha\in\mathcal{A}$ which refines $\mathcal{V}$. Let $\mu=\{\alpha_{1},\ldots,\alpha_{n},\alpha\}\geq\lambda$. Then $\operatorname{cov}_{\mu}$ refines the cover $\alpha$ and hence $\mathcal{U}$. Let $v_{1},\ldots,v_{m}$ be the vertices of an simplex of $F_{\mu}$. Then $\wedge v_{i}\cap\wedge v_{j}\not=\emptyset$, $i,j=1,\ldots,m$. Since each $\wedge v_{i}$ is contained in some $U\in\mathcal{U}$ and $\mathcal{U}$ is a star refinement of the cover $\{U_{x}\mid x\in X\}$, we find a point $x\in X$ such that $U_{x}$ contains all $\wedge v_{i},~{}i=1,\ldots m$. Thus $x\in\wedge\pi^{\mu}_{\lambda}(v_{i})$ for $i=1,\ldots,m$ and $\{\pi_{\lambda}^{\mu}(v_{1}),\ldots,\pi_{\lambda}^{\mu}(v_{m})\}$ spans a simplex of $N_{\lambda}$. Thus $\pi_{\lambda}^{\mu}(F_{\mu})\subset N_{\lambda}$. The last statement follows from the first and Morita’s Lemma [8, Chap. II, Section 2,2, Theorem 5]. ∎

2.2. Polyhedral expansions

Every polyhedral resolution ([8]) of a space $X$ gives an HPol-expansion ([8]) defining the shape type of $X$. Likewise the above inverse systems $(N_{\lambda},\pi_{\lambda}^{\mu};\Lambda)$ and $(F_{\lambda},\pi_{\lambda}^{\mu};\Lambda)$ yield polyhedral expansions of spaces under some mild assumption on the collection $\mathcal{A}$. The expansions so obtained have two features: (i) the polyhedra $N_{\lambda}$ or $F_{\lambda}$ have specific triangulations and the bonding maps are simplicial with respect to these triangulations, while the resolution constructed in [8, Chap.1, Section 6.4, Theorem 7] has continuous, but not necessarily simplicial, bonding maps. (ii) for the systems $(N_{\lambda},\pi_{\lambda}^{\mu};\Lambda)$ and $(F_{\lambda},\pi_{\lambda}^{\mu};\Lambda)$, the equality $\pi_{\lambda}^{\nu}=\pi_{\lambda}^{\mu}\circ\pi_{\mu}^{\nu}$ holds for $\lambda\leq\mu\leq\nu$, while in the classical Čech system on the basis of refinement-relation, $\pi_{\lambda}^{\nu}$ is only homotopic to $\pi_{\lambda}^{\mu}\circ\pi_{\mu}^{\nu}$.

Here, rather than recalling the notion of HPol-expansions, we follow the formulation of [10] (see also [8, Chap.I, section 2.4]). For spaces $Y$ and $K$, $[Y,K]$ denotes the set of homotopy classes of maps $Y\to K$. Every continuous map $g:X\to Y$ induces a function $g^{\#}:[Y,K]\to[X,K]$. The inverse system $(N_{\lambda},\pi_{\lambda}^{\mu};\Lambda)$ yields a direct system $([N_{\lambda},K],(\pi_{\lambda}^{\mu})^{\#};\Lambda)$ of sets which defines the direct limit(or colimit) $\varinjlim([N_{\lambda},K],(\pi_{\lambda}^{\mu})^{\#};\Lambda)$ in the category Sets of sets and functions. The above map $p_{\lambda}:X\to N_{\lambda}$ induces a function $p^{\#}_{\lambda}:[N_{\lambda},K]\to[X,K]$ for each $\lambda$ which yields the limit map $p^{\sharp}:\varinjlim([N_{\lambda},K],(\pi_{\lambda}^{\mu})^{\#};\Lambda)\rightarrow[X,K]$. In the sequel, $K$ is assumed to be a simplicial complex with the weak topology. Since the classes of spaces that are homotopy equivalent to

• (i)

simplicial complexes with the weak topology,

• (ii)

simplicial complexes with the metric topology,

• (iii)

CW complexes,

• (iv)

metric ANR’s,

all coincide ([9], [8] etc), we may assume that $K$ belong to any of the above classes. The system $(p_{\lambda}:X\to N_{\lambda})$ is an HPol-expansion of $X$ (see [8]), or a system satisfying (M1) and (M2) of [7, p.256], precisely when the above $p^{\#}$ is a bijection for each simplicial complex $K$. Shape theory for general topological spaces was studied in [11] where it was shown that $\text{Sh}(X)=\text{Sh}(\tau X)=\text{Sh}(\mu X)$ where $\tau$ denotes the Tychonoff functor [10] and $\mu X$ denotes the completion with respect to the finest uniformity of $\tau X$. Thus there is no loss of generality in assuming that the spaces are topologically complete in the next results.

Proposition 2.7**.**

Let $X$ be a topologically complete space and $\mathcal{A}$ be a cofinal subfamily of all closed, locally finite, normal covers of $X$. Then $\mathcal{A}$ satisfies the conditions I) and II). Further we have a bijection

[TABLE]

for each simplicial complex $K$. If $X$ is a paracompact space, then

[TABLE]

is also a bijection for each simplicial complex $K$.

Proof.

First notice that the family of locally finite open normal covers is cofinal in the family of normal covers. Indeed as was shown in §2 of [10], for an open normal cover $\mathcal{U}$ in $X$ there exist a continuous map $f:X\rightarrow Y$ to a metric space $Y$ and an open cover $\mathcal{V}$ of $Y$ such that the cover $f^{-1}(\mathcal{V})$ refines $\mathcal{U}$ (see also [8, Chap.I, section 6.2, Lemma 1]). By the paracompactness of $Y$, there exists a locally finite open cover $\mathcal{W}$ of $Y$ that refines $\mathcal{V}$. Then $f^{-1}(\mathcal{W})$ is a locally finite, open, normal cover of $X$ refining cover $\mathcal{U}$.

We show that $\mathcal{A}$ satisfies the conditions I) and II). Let $U\subset X$ be an open set and $x\in U$. There exists a continuous function $\phi:X\rightarrow[0,1]$ such that $\phi(x)=1$, $\mathrm{supp}(\phi):=\mathrm{cl}\phi^{-1}((0,1])\subset U$, and $\phi\equiv 1$ on a neighborhood of $x$. Therefore $\{U,X\setminus\{x\}\}$ is a normal cover with partition of unity $\{\phi,1-\phi\}$. Take a normal open cover $\mathcal{U}$ which is a star refinement of $\{U,X\setminus\{x\}\}$. By the cofinality of $\mathcal{A}$ and the remark at the beginning of the proof, there exists an $\alpha\in\mathcal{A}$ which refines $\mathcal{U}$, and thus $\alpha$ is a star refinement of $\{U,X\setminus\{x\}\}$. This implies the condition I). To verify the condition II), let $\{f(\alpha)~{}|~{}\alpha\in\mathcal{A}\}$ be a family of subsets with the finite intersection property such that $f(\alpha)\in\alpha$ for each $\alpha\in\mathcal{A}$. By the cofinality of $\mathcal{A}$ and the cofinality of the family of the locally finite normal open coverings in that of the open normal coverings, the collection $\{f(\alpha)~{}|~{}\alpha\in{\mathcal{A}}\}$ contains arbitrarily small sets with respect to normal open covers. By recalling that the finest uniformity is generated by the family of normal open covers of $X$ ([5, Chap.8, 8.1.C]), we conclude that $\bigcap f(\alpha)\neq\emptyset$ by the topological completeness of $X$.

The bijection (2.5) is a consequence of [10, Theorem 4.3]. Notice, that the use of open/closed covers causes no essential difference since each of the family of closed/open covers involved is cofinal in the other. Notice also that the nerve complex $\mathcal{N}(\operatorname{cov}_{\lambda})$ may be replaced with the complex $N_{\lambda}$ because they are homotopically equivalent by a natural homotopy equivalence $N_{\lambda}\to\mathcal{N}(\operatorname{cov}_{\lambda})$.

If $X$ is a paracompact then the second conclusion (2.6) follows from Theorem 2.6. ∎

Note that a closed cover $\mathcal{G}$ of a paracompact space $X$ with $\bigcup\{\mathrm{int}G~{}|~{}G\in{\mathcal{G}}\}=X$ is a normal cover.

A inverse system $\mathbf{X}$ of polyhedra with continuous bonding maps may fail to be an HPol-expansion of the limit space $\varprojlim{\mathbf{X}}$ when the limit space is not compact ([8, Chap. I, Section 6, Example 1]). It follows from Proposition 2.7 that the inverse systems ${\mathbf{N}}_{\mathcal{A}}$ and ${\mathbf{F}}_{\mathcal{A}}$ constructed above are HPol-expansions of the limit space $N_{\infty}=F_{\infty}$ under a suitable assumption on the space $X$ and the family $\mathcal{A}$.

Corollary 2.8**.**

Let $X$ be a topologically complete space and let $\mathcal{A}$ be a family of closed covers as in Proposition 2.7. Then we have a bijection

[TABLE]

for each simiplicial complex $K$. If moreover $X$ is paracompact, then the same holds for $(F_{\lambda},\pi_{\lambda}^{\mu};\Lambda)$.

For compact spaces we obtain the same conclusion under a weaker hypothesis.

Proposition 2.9**.**

If $X$ is a compact Hausdorff space and $\mathcal{A}$ is a family satisfying the condition I), then $\mathcal{A}$ is cofinal in the family of all closed, locally finite, normal covers of $X$. Thus we obtain the bijections of Proposition 2.7. In particular $\bf{N}_{\mathcal{A}}$ and $\bf{F}_{\mathcal{A}}$ are HPol-expansions of the limit space $N_{\infty}=F_{\infty}$.

Proof.

We show the first statement. The rest follows from Proposition 2.7. For a finite closed normal cover $\alpha$ of $X$, we show that there is $\lambda\in\Lambda$ such that, for each $x\in X$, $\mathrm{star}_{\lambda}(x)\subset\mathrm{int}F$ for some $F\in\alpha$. This implies that $\lambda$ refines $\alpha$. Suppose that $F_{\lambda}=\{x\in X\mid\bigcup\mathrm{star}_{\lambda}(x)\not\subset\mathrm{int}(F)~{}\forall F\in\alpha\}\neq\emptyset$ for each $\lambda$. Each $F_{\lambda}$ is a closed set and $F_{\mu}\subset F_{\lambda}$ for $\mu\geq\lambda$. Then $\bigcap F_{\lambda}\not=\emptyset$ and take a point $x\in\bigcap F_{\lambda}$. Thus we have $x\in\operatorname{int}(F)$ for some $F\in\alpha$. Then, by the condition I), we obtain $\bigcup\mathrm{star}_{\alpha}(x)\subset\operatorname{int}(F)$ for some $\lambda\in\Lambda$, a contradiction. ∎

3. Cell structures and polyhedral expansions

A graph is an ordered pair $(G,r)$ of a (possibly infinite) discrete set $G$ and a symmetric and reflexive binary relation $r$ on $G$. Two elements $a,b\in G$ are said to be adjacent if $(a,b)\in r$. For $a\in G$ let $\text{star}_{r}(a)=\{b\in G|(a,b)\in r\}$. A graph homomorphism $f:(G,r)\rightarrow(G^{\prime},r^{\prime})$ between graphs, that is a function $f:G\to G^{\prime}$ satisfying $(a,b)\in r$ implies $(f(a),f(b))\in r^{\prime}$, is termed as a continuous map here. For an inverse system of graphs $\mathbf{G}=(G_{\lambda},\pi_{\lambda}^{\mu};\Lambda)$ we continue to use the notation of Section 2: let $G_{\infty}=\varprojlim(G_{\lambda},\pi_{\lambda}^{\mu};\Lambda)$ and for $\lambda$, let $\pi_{\lambda}:G_{\infty}\rightarrow G_{\lambda}$ be the $\lambda^{th}$ coordinate projection. For a point $z\in G_{\infty}$, the element $\pi_{\lambda}(z)\in G_{\lambda}$ is also denoted by $z(\lambda)$. A cell-structure $\mathbf{G}=(G_{\lambda},\pi^{\mu}_{\lambda}:G_{\mu}\to G_{\lambda};\Lambda)$ is an inverse system of graphs and continuous bonding maps satisfying the conditions a) and b) below.

1. a)

for each $z\in G_{\infty}$ and $\lambda\in\Lambda$, there exists $\mu\geq\lambda$ such that

[TABLE] 2. b)

for each $z\in G_{\infty}$ and $\lambda\in\Lambda$, there exists $\mu\geq\lambda$ such that $\pi^{\mu}_{\lambda}(\text{star}_{r_{\mu}}(z(\mu)))$ is finite.

A point of $G_{\infty}$ is called a thread. Two threads $z,z^{\prime}\in G_{\infty}$ are equivalent, written as $z\sim z^{\prime}$, if $z(\lambda)$ and $z^{\prime}(\lambda)$ are adjacent in $G_{\lambda}$ for each $\lambda$. By [3, Theorem 3.4], the relation $\sim$ is a closed equivalence relation with compact equivalence classes and the quotient space $G^{\ast}=G_{\infty}/\sim$ with the quotient topology induced by $G_{\infty}$, is called the topological space represented by the cell structure ${\mathbf{G}}=(G_{\lambda},\pi^{\mu}_{\lambda};\Lambda)$. The quotient projection is denoted by $\pi_{\mathbf{G}}:G_{\infty}\to G^{\ast}$. It is a perfect map because it is a closed map with fibers being compact equivalence classes. A net $y:\Lambda\rightarrow\bigcup_{\lambda}G_{\lambda}$ indexed by $\Lambda$ with $y(\lambda)\in G_{\lambda}$ for each $\lambda$ is called a Cauchy net if there exists $\lambda\in\Lambda$ such that, for each $\lambda^{0}$, we have $\pi^{\lambda^{1}}_{\lambda^{0}}(y(\lambda^{1}))$ and $\pi^{\lambda^{2}}_{\lambda^{0}}(y(\lambda^{2}))$ are adjacent for all $\lambda^{1},\lambda^{2}\geq\lambda,\lambda^{0}$. A Cauchy net $y$ is said to converge to a thread $z\in G_{\infty}$ if there exists $\lambda\in\Lambda$ such that $y(\mu)$ and $z(\mu)$ are adjacent for each $\mu\geq\lambda$. The cell structure is complete if every Cauchy net converges to a thread.

The construction given in Section 2 yields cell structures associated with topologically complete spaces. Let $X$ be a topologically complete space and let $\mathcal{A}$ be a set of closed, locally finite, normal covers of $X$ which satisfies the conditions I)-II) of Section 2. Under the notation of Section 2, the set of vertices $N^{(0)}_{\lambda}=F^{(0)}_{\lambda}$, a discrete set, admits the symmetric and reflexive relation $r_{\lambda}$ defined by $(u,v)\in r_{\lambda}$, $u,v\in F_{\lambda}^{(0)}$, if and only if $\wedge u\cap\wedge v\neq\emptyset$, or equivalently $u$ and $v$ span an 1-simplex of $F_{\lambda}$. Thus the graph $(F^{(0)}_{\lambda},r_{\lambda})$ is isomorphic to the 1-skeleton of $F_{\lambda}$. The next theorem shows that the inverse system $\mathbf{F}^{(0)}=(F^{(0)}_{\lambda},\pi^{\mu}_{\lambda},\Lambda)$ defined in Section 2, with the above relation $r_{\lambda}$ for each $\lambda\in\Lambda$, is a cell structure. The space $F_{\infty}^{(0)}=\varprojlim\mathbf{F}^{(0)}$ with the quotient projection $\pi_{{\mathbf{F}}^{(0)}}:F_{\infty}^{(0)}\to F_{\infty}^{(0)}/\sim$ represent the space $X$ in a canonical way. Part of this theorem was first proved in [3, Theorem 8.1].

Theorem 3.1**.**

Let $X$ be a topologically complete space and let $\mathcal{A}$ be a cofinal subfamily of the family of all closed, locally finite, normal covers of $X$. Then the inverse system $\mathbf{F}^{(0)}=(F^{(0)}_{\lambda},\pi^{\lambda^{\prime}}_{\lambda},\Lambda)$ is a complete cell structure representing a space canonically homeomorphic to $X$: there exists a unique homeomorphism $h:X\to{(F^{(0)})}^{\ast}:=F_{\infty}^{(0)}/\sim$ from $X$ to the quotient space ${(F^{(0)})}^{\ast}$ defined by ${\mathbf{F}}^{(0)}$ such that $h\circ\pi|F_{\infty}^{(0)}=\pi_{{\mathbf{F}}^{(0)}}$.

Proof.

Recall that $\mathcal{A}$ satisfies the conditions I) and II) by Proposition 2.7. We show that $\mathbf{F}^{(0)}$ is a cell structure. For this, let $z\in F^{(0)}_{\infty}$. By the definition of $\pi$ we have $\bigcap_{\lambda\in\Lambda}z(\lambda)=\{\pi(z)\}$. For $\lambda\in\Lambda$, let $V=X\setminus\bigcup\{U\in\operatorname{cov}_{\lambda}|\pi(z)\notin U\}$. Then $V$ is an open neighborhood of $\pi(z)$ and meets only finitely many elements of $\operatorname{cov}_{\lambda}$. By Lemma 2.2 (1), there exists $\lambda^{\prime}\geq\lambda$ so $\text{star}^{3}_{\operatorname{cov}_{\lambda^{\prime}}}(\pi(z))\subset V$. Then $\pi^{\lambda^{\prime}}_{\lambda}(\text{star}^{2}_{\operatorname{cov}_{\lambda^{\prime}}}(z(\lambda^{\prime})))\subset\pi^{\lambda^{\prime}}_{\lambda}(\text{star}^{3}_{\operatorname{cov}_{\lambda}}(\pi(z)))\subset\text{star}_{\operatorname{cov}_{\lambda}}(z(\lambda))$. Thus $\mathbf{F}^{(0)}=(F^{(0)}_{\lambda},\pi^{\mu}_{\lambda},\Lambda)$ satisfies the conditions a) and b) and is a cell structure.

In order to prove the completeness, let $(y(\lambda))_{\lambda\in\Lambda}$ be a Cauchy net with $y(\lambda)\in F_{\lambda}^{(0)}(\lambda\in\Lambda)$. We show

[TABLE]

which implies that $(y(\lambda))_{\lambda}$ converges to a thread by [3, Lemma 3.7]. Take $\lambda^{-1}\in\Lambda$ such that for each $\lambda^{0},\lambda^{1},\lambda^{2}$ with $\lambda^{1},\lambda^{2}\geq\lambda^{0},\lambda^{-1}$ we have

[TABLE]

For $\lambda\in\Lambda$, let $F_{\lambda}=\operatorname{cl}(\bigcup_{\mu\geq\lambda}\wedge y(\mu))$. The family $\{F_{\lambda}\}$ has the finite intersection property. We first verify

[TABLE]

Indeed, for a locally finite open normal cover $\mathcal{U}$ take an open normal cover $\mathcal{V}$ which $\mathrm{star}^{2}$-refines $\mathcal{U}$. By taking a closed shrinking and by using the cofinality of $\mathcal{A}$, we have an $\alpha\in{\mathcal{A}}$ that refines $\mathcal{V}$. Take $\lambda\geq\{\alpha\},\lambda^{-1}$. Since $\operatorname{cov}_{\lambda}$ refines $\mathcal{V}$, there exists $V\in\mathcal{V}$ such that $\wedge y(\lambda)\subset V$. For each $\mu\geq\lambda$, we see, by (3.1), $\wedge\pi_{\lambda}^{\mu}(y(\mu))\cap\wedge y(\lambda)\neq\emptyset$. Hence we have $\wedge y(\mu)\subset\wedge\pi_{\lambda}^{\mu}(y(\mu))\subset\bigcup\mathrm{star}_{\operatorname{cov}_{\lambda}}(\wedge y(\lambda))\subset\mathrm{star}_{\mathcal{V}}(V)$ and thus $F_{\lambda}\subset\mathrm{star}_{\mathcal{V}}^{2}(V)\subset U$ for some $U\in\mathcal{U}$. This proves $(\ast\ast)$.

Recall that the collection of the normal open covers forms a base of the finest uniformity of the Tychnoff space $X$. The topological completeness of $X$ together with $(\ast\ast)$ then imply that the family $\{F_{\lambda}\}$ contains arbitrarily small sets with respect to the finest uniformity. This together with the finite intersection property imply that the intersection $\bigcap_{\lambda\in\Lambda}F_{\lambda}$ contains a point $x$. For the proof of $(\ast)$, take an arbitrary $\lambda\in\Lambda$ and choose an open neighborhood $U$ of $x$ such that $\mathrm{star}_{\operatorname{cov}_{\lambda}}(U)$ is a finite set by the local finiteness of $\operatorname{cov}_{\lambda}$. Applying $(\ast\ast)$ to the cover $\{U,X\setminus\{x\}\}$ and noticing $x\in\bigcap F_{\nu}$, we find $\mu\geq\lambda$ such that $F_{\mu}\subset U$. For each $\nu\geq\mu$ we have $\wedge y(\nu)\subset F_{\mu}\subset U$ which implies $\wedge\pi_{\lambda}^{\nu}(y(\nu))\cap U\neq\emptyset.$ Thus we have $A_{\lambda,\mu}\subset\mathrm{star}_{\operatorname{cov}_{\lambda}}(U)$ and $A_{\lambda,\mu}$ is a finite set. This proves $(\ast)$ and proves the completeness of ${\mathbf{F}}^{(0)}$.

The last statement follows from the definitions of the relation $\sim$, the maps $\pi$(see also Proposition 2.4 (2)) and $\pi_{{\mathbf{F}}^{(0)}}$. Recall that $\pi_{{\mathbf{F}}^{(0)}}$ is a perfect map and note that $\pi|_{N^{0}_{\infty}}:N^{(0)}_{\infty}\rightarrow X$ is also a perfect map since $N^{(0)}_{\infty}$ is a closed subset of $N_{\infty}$. ∎

For a graph $(G,r)$, we define a flag complex ${\mathcal{F}}(G)$ as follows: the vertex set of ${\mathcal{F}}(G)$ is the set of vertices of $G$; a set of vertices $\{v_{1},\ldots,v_{n}\}$ spans a simplex of ${\mathcal{F}}(G)$ if $v_{i},v_{j}$ are adjacent in $G$ for each $i,j=1,\ldots,n$. Every graph homomorphism $f:G\to H$ induces a simplicial map ${\mathcal{F}}(f):{\mathcal{F}}(G)\to{\mathcal{F}}(H)$.

Remark 3.2**.**

A cell structure ${\mathbf{G}}=(G_{\lambda},\pi_{\lambda}^{\mu},\Lambda)$ that represents a space $Y$ induces an inverse system ${\mathcal{F}}({\mathbf{G}})=({\mathcal{F}}(G_{\lambda}),{\mathcal{F}}(\pi_{\lambda}^{\mu}),\Lambda)$ of flag complexes. Also the equivalence relation $\sim$ on $G_{\infty}$ naturally extends to the one on $\varprojlim{\mathcal{F}}({\mathbf{G}})$ in such a way that the quotient space $\varprojlim{\mathcal{F}}({\mathbf{G}})/\sim$ is homeomorphic to $Y$. Applying this to the cell structure ${\mathbf{F}}^{(0)}$ of Theorem 3.1 with a suitable family $\mathcal{A}$, we see that the system ${\mathcal{F}}({\mathbf{F}}^{(0)})$ is precisely the inverse system ${\mathbf{F}}_{\mathcal{A}}=(F_{\lambda},\pi_{\lambda}^{\mu};\Lambda)$ constructed in Section 2, and the quotient map $\varprojlim{\mathcal{F}}({\mathbf{F}}^{(0)})\to\varprojlim{\mathcal{F}}({\mathbf{F}}^{(0)})/\sim$ is identified with the homotopy equivalence $\pi:F_{\infty}\to X$. In particular, Proposition 2.7, Corollary 2.8 and Proposition 2.9 hold for the system ${\mathcal{F}}({\mathbf{F}}^{0})$. Thus a cell structure $\mathbf{G}$ of a paracompact space $X$ with a suitable family of closed normal covers $\mathcal{A}$ yields a polyhedral expansion of both of $X$ and the limit space $\varprojlim{\mathcal{F}}({\mathbf{G}})$.

Question 3.3**.**

For what class of space $X$ and for which cell structure $\mathbf{G}$ representing $X$ do we have a polyhedral expansion $\varprojlim{\mathcal{F}}({\mathbf{G}})\to{\mathcal{F}}({\mathbf{G}})$ with a shape/homotopy equivalence $\varprojlim{\mathcal{F}}({\mathbf{G}})\to X$?

For a space $X$ for which the above question has a positive answer, we would have a way to study the shape type of $X$ by means of cell structures representing $X$.