On the cardinality of $\pi(\delta)$

TL;DR

This paper investigates the size and structure of transitive quasi-uniformities within a quasi-proximity class, establishing conditions for their cardinality and uniqueness.

Contribution

It proves the lower bound on the cardinality of transitive quasi-uniformities and characterizes their transitive elements under certain conditions.

Findings

Cardinality of transitive quasi-uniformities is at least $2^{2^{eth_0}}$ when multiple exist.

Characterization of transitive elements of $ ext{pi}( extdelta)$ when ${ extcal V}_{ extdelta}$ is transitive.

Conditions for the uniqueness of transitive quasi-uniformity in $ ext{pi}( extdelta)$.

Abstract

We prove that the cardinality of transitive quasi-uniformities in a quasi-proximity class is at least $2^{2^{\aleph_0}}$ if there exist at least two transitive quasi-uniformities in the class. The transitive elements of $\pi(\delta)$ are characterized if ${\cal V}_{\delta}$ is transitive, and in this case we give a condition when there exists a unique transitive quasi-uniformity in $\pi(\delta)$.

Equations

$$\bigcap _{i=1}^n{U}_{M_i}\cap U_j\subseteq U_N$$

$$N=\bigcup _{x\in N}(U_{\{M_1,\dots ,M_n\}}\cap U_j)(x)=$$

$$\bigcup _{x:U^{\prime }(x)=M_1}(M_1\cap U_j(x))\bigcup \bigcup _{x:U^{\prime }(x)=M_2}(M_2\cap U_j(x))\bigcup \dots \bigcup \bigcup _{x:U^{\prime }(x)=M_n}(M_n\cap U_j(x))=$$

$$\bigl{(}M_{1}\cap U_{j}(\{x:U^{\prime}(x)=M_{1}\})\bigr{)}\bigcup\ldots\bigcup\bigl{(}M_{n}\cap U_{j}(\{x:U^{\prime}(x)=M_{n}\})\bigr{)}\in\mbox{${{{\cal B}}}$}$$

$${\alpha }_A=\{X,N_i:i\notin A\}.$$

$$\mbox{${{{\cal V}}}$}_{\sigma}=\mbox{\rm fil${}_{X\times X}$}\{\mbox{$\mbox{${{{\cal V}}}$}_{\delta}$},U_{A}:A\in\sigma\}.$$

$${\alpha }_A=\{X,\bigcap _{j=1}^{\infty }{N}_j\}\bigcup \{N_i:i\notin A\}.$$

$$\mbox{${{{\cal V}}}$}_{\sigma}=\mbox{\rm fil${}_{X\times X}$}\{\mbox{$\mbox{${{{\cal V}}}$}_{\delta}$},U_{A}:A\in\sigma\}.$$

Full text

Keywords: compatible quasi-proximity, quasi-proximity class, compatible quasi-uniformity, transitive quasi-uniformity, totally bounded quasi-uniformity, l-base, p-filter

AMS Subject Classification: 54E15; 54A25

1 Introduction

The purpose of this paper is the generalization of a result of [12] which states that there exist either a unique or at least $2^{2^{\aleph_{0}}}$ compatible transitive quasi-uniformities on a topological space. Moreover we proved there that $|\pi(\delta^{1})|=1$ or \geq\mbox{2^{2^{\aleph_{0}}}} where $\delta^{1}$ denotes the finest compatible quasi-proximity on $X$, and $\pi(\delta^{1})$ is its quasi-proximity class, namely \pi(\delta^{1})=\{\mbox{{{{\cal V}}}}:\mbox{{{{\cal V}}}}\supset\mbox{{{{\cal P}}}}\} where ${{{\cal P}}}$ is the Pervin quasi-uniformity of $X$. The natural question arises : is something similar true for $\pi(\delta)$ if $\delta$ is an arbitrary compatible quasi-proximity? In this paper we partly answer this question by proving that |\mbox{\pi(\delta)}|=1 or \geq\mbox{2^{2^{\aleph_{0}}}} if the coarsest element of $\pi(\delta)$ is transitive.

In [13] we prove that in the class of infinite locally compact T2 spaces $|\pi(\delta^{0})|=1$ if and only if $X$ is compact or non-Lindelöf ($\delta^{0}$ denotes the coarsest compatible quasi-proximity) and if $X$ is non-compact and Lindelöf, then |\pi(\delta^{0})|\geq\mbox{2^{2^{\aleph_{0}}}}.

We give two elementary definitions.

Definition 1.1

Let $(X,\tau)$ be a topological space. Then $N(X)$ or $N(\tau)$ ($T(X)$ or $T(\tau)$) denotes the set of all compatible (transitive) quasi-uniformities on $X$ respectively.

Definition 1.2

A base ${{{\cal B}}}$ of a topological space is called an l-base or a lattice-base if it is closed under finite union and finite intersection and \mbox{\emptyset},X\in\mbox{{{{\cal B}}}}.

We enumerate some results connected with this notion. For the proofs of these results the reader may wish to consult [11].

We say that $\alpha$ is an interior preserving open cover if it is an open cover and for every $x\in X$ the set $\bigcap\{N\in\alpha:x\in N\}$ is open or equivalently $\alpha^{\prime}\subseteq\alpha{{\ \rm implies\ }}\bigcap\alpha^{\prime}$ is open. We can assign a transitive neighbournet $U_{\alpha}$ to $\alpha$ in the following way: \mbox{U_{\alpha}}(x)=\bigcap\{N\in\alpha:x\in N\}\ (x\in X). If \alpha^{\prime}=\{\mbox{U_{\alpha}}(x):x\in X\}{{\ \rm then\ }}\alpha^{\prime} is also an interior preserving open cover and U_{\alpha^{\prime}}=\mbox{U_{\alpha}}. An important remark is that if $\alpha$ is finite then \{\mbox{U_{\alpha}}(x):x\in X\} is finite too and if \mbox{U_{\alpha}}\in\mbox{{{{\cal P}}}}{{\ \rm then\ }}\alpha is finite where ${{{\cal P}}}$ denotes the Pervin quasi-uniformity of $X$ namely \mbox{{{{\cal P}}}}=\mbox{\rm fil{}{X\times X}}\{\mbox{U{\alpha}}:\alpha is finite$\}$. It is known that a quasi-uniformity is totally bounded if and only if it is contained in ${{{\cal P}}}$ (see [2]1).

In [11] we showed that there is a one-to-one correspondence between the set of compatible totally bounded transitive quasi-uniformities and the set of l-bases. Namely if ${{{\cal V}}}$ is a quasi-uniformity of the mentioned type then \mbox{{{{\cal B}}}}(\mbox{{{{\cal V}}}})=\{N\in\tau:U_{N}\in\mbox{{{{\cal V}}}}\} is an l-base, and if ${{{\cal B}}}$ is an l-base then \mbox{{{{\cal V}}}}(\mbox{{{{\cal B}}}})=\mbox{\rm fil{}_{X\times X}}\{U_{N}:N\in\mbox{{{{\cal B}}}}\} is a totally bounded transitive quasi-uniformity, where $U_{N}=(N\times N)\cup((X-N)\times X)=U_{\{N,X\}}$ for the cover$\{N,X\}$.

If we say that $\delta$ is a quasi-proximity we use it in the sense of [2]1.22. If ${{{\cal V}}}$ is a quasi-uniformity, then \delta(\mbox{{{{\cal V}}}}){{\ \rm and\ }}\tau(\mbox{{{{\cal V}}}}) will always denote the quasi-proximity and the topology induced by ${{{\cal V}}}$ respectively, $\tau(\delta)$ will denote the topology induced by $\delta$. Let \mbox{\pi(\delta)}=\{\mbox{{{{\cal V}}}}:\delta(\mbox{{{{\cal V}}}})=\delta\}. We know from [2]1.33 that  for every \delta,\ \mbox{\pi(\delta)}\not=\mbox{\emptyset}. Moreover there exists a coarsest element of $\pi(\delta)$, it is denoted by \mbox{{{{\cal V}}}}_{\delta}. This is totally bounded and the only totally bounded member of $\pi(\delta)$. If a quasi-uniformity ${{{\cal V}}}$ is given then \mbox{{{{\cal V}}}}_{\omega} denotes the coarsest element of \pi\bigl{(}\delta(\mbox{{{{\cal V}}}})\bigr{)}.

2 The main results

First we want to characterize the transitive elements of $\pi(\delta)$ where $\delta$ is a compatible quasi-proximity such that \mbox{{{{\cal V}}}}_{\delta} is transitive. To this aim we will need some lemmas.

Lemma 2.1

Let $\delta$ be a quasi-proximity on $X$ compatible with $\tau$ such that \mbox{{{{\cal V}}}}_{\delta} is transitive. Let \mbox{{{{\cal V}}}}\in T(X)\ such\ that\ \mbox{\mbox{${{{\cal V}}}$}{\delta}}\subseteq\mbox{{{{\cal V}}}}. In this case \mbox{{{{\cal V}}}}\in\mbox{\pi(\delta)} if and only if N\in\tau,\ U_{N}\in\mbox{{{{\cal V}}}} imply that N\in\mbox{{{{\cal B}}}}(\delta)=\mbox{{{{\cal B}}}}(\mbox{\mbox{${{{\cal V}}}$}{\delta}}).

Proof: By [10]2.6 we know that (\mbox{\mbox{${{{\cal V}}}$}_{\omega}})_{t}=(\mbox{{{{\cal P}}}})_{t}\cap(\mbox{{{{\cal V}}}})_{t} where for a quasi-uniformity ${{{\cal U}}}$, (\mbox{{{{\cal U}}}})_{t}=\{U\in\mbox{{{{\cal U}}}}:U is transitive$\}$.

If \mbox{{{{\cal V}}}}\in\mbox{\pi(\delta)} then \mbox{\mbox{${{{\cal V}}}$}{\omega}}=\mbox{\mbox{${{{\cal V}}}$}{\delta}}. By the previous observation if U_{N}\in\mbox{{{{\cal V}}}}{{\ \rm then\ }}U_{N}\in\mbox{\mbox{${{{\cal V}}}$}{\delta}}{{\ \rm hence\ }}N\in\mbox{{{{\cal B}}}}(\delta). Let us prove the sufficiency. We get (\mbox{{{{\cal V}}}})_{t}\cap(\mbox{{{{\cal P}}}})_{t}\subseteq\mbox{\mbox{${{{\cal V}}}$}{\delta}}{{\ \rm and\ }}\mbox{\mbox{${{{\cal V}}}$}{\omega}}\subseteq\mbox{\mbox{${{{\cal V}}}$}{\delta}} which yields that \mbox{{{{\cal V}}}}\in\mbox{\pi(\delta)}. $\bullet$

Lemma 2.2

Let \mbox{{{{\cal V}}}}\in\mbox{T(\tau)},\ \mbox{{{{\cal B}}}}=\mbox{{{{\cal B}}}}(\mbox{\mbox{${{{\cal V}}}$}_{\omega}})\ and\ U\in\mbox{{{{\cal V}}}} be transitive. In this case U(A)\in\mbox{{{{\cal B}}}} if $A\subseteq X$.

Proof: Let $A\subseteq X{{\ \rm and\ }}N=U(A)$. It is easy to check that $U\subseteq U_{N}$ so U_{N}\in\mbox{{{{\cal V}}}},{{\ \rm and\ }}U_{N}\in\mbox{\mbox{${{{\cal V}}}$}_{\omega}}. $\bullet$

Corollary 2.3

Let \mbox{{{{\cal V}}}}\in\mbox{T(\tau)},\ \mbox{{{{\cal B}}}}=\mbox{{{{\cal B}}}}(\mbox{\mbox{${{{\cal V}}}$}{\omega}})\ and\ \mbox{U{\alpha}}\in\mbox{{{{\cal V}}}} where $\alpha$ is an interior preserving open cover. Then $A\in\alpha$ implies that A\in\mbox{{{{\cal B}}}}.

Proof: A=\mbox{U_{\alpha}}(A). $\bullet$

Proposition 2.4

Let $\delta$ be a compatible quasi-proximity on $X$ such that \mbox{{{{\cal V}}}}_{\delta} is transitive and let ${{{\cal B}}}$ be the l-base associated with \mbox{{{{\cal V}}}}_{\delta}. Let \mbox{{{{\cal V}}}}=\mbox{\rm fil{}{X\times X}}\{\mbox{\mbox{${{{\cal V}}}$}{\delta}},U_{i}:i\in I\} where $U_{i}$ is transitive ($i\in I$) and the system $\{U_{i}:i\in I\}$ is closed under finite intersection. Then \mbox{{{{\cal V}}}}\in\mbox{\pi(\delta)} if and only if \forall i\in I,\ \forall A\subseteq X\ U_{i}(A)\in\mbox{{{{\cal B}}}}.

Proof: If \mbox{{{{\cal V}}}}\in\mbox{\pi(\delta)}{{\ \rm then\ }}\mbox{\mbox{${{{\cal V}}}$}{\omega}}=\mbox{\mbox{${{{\cal V}}}$}{\delta}} and we get the statement by the previous lemma (2.2).

To prove the opposite case let N\in\tau=\tau(\delta),\ U_{N}\in\mbox{{{{\cal V}}}}. We need that N\in\mbox{{{{\cal B}}}} by 2.1. We know that there are M_{i}\in\tau,\ U_{M_{i}}\in\mbox{\mbox{${{{\cal V}}}$}_{\delta}}{{\ \rm and\ }}j\in I{{\ \rm such\ that\ }}

[TABLE]

or in other words $U_{\{M_{1},\ldots,M_{n},X\}}\cap U_{j}\subseteq U_{N}$ for the cover $\{M_{1},\ldots,M_{N},X\}$. Obviously M_{i}\in\mbox{{{{\cal B}}}} and we can assume the system $\{M_{1},\ldots,M_{n},X\}$ is closed for union and intersection and $M_{n}=X$. Let $U^{\prime}=U_{\{M_{1},\ldots,M_{n}\}}$. If $x\in N{{\ \rm then\ }}{{\ \rm there\ is\ }}M_{i}{{\ \rm such\ that\ }}x\in M_{i}\cap U_{j}(x)\subseteq U_{N}(x)=N$. Now

[TABLE]

[TABLE]

[TABLE]

since U_{j}(\{x\in N:U^{\prime}(x)=M_{k}\})\in\mbox{{{{\cal B}}}} by assumption and ${{{\cal B}}}$ is an l-base. $\bullet$

Corollary 2.5

Let $\delta$ be a quasi-proximity, \mbox{{{{\cal V}}}}_{\delta} be transitive, ${{{\cal B}}}$ be the l-base associated with it and ${{{\cal V}}}$ be a transitive quasi-uniformity on $X$.Then \mbox{{{{\cal V}}}}\in\mbox{\pi(\delta)} if and only if \mbox{\mbox{${{{\cal V}}}$}_{\delta}}\subseteq\mbox{{{{\cal V}}}}\ and\for every transitive U\in\mbox{{{{\cal V}}}} and for every A\subseteq X,\ U(A)\in\mbox{{{{\cal B}}}}.

Proof: This is an obvious consequence of 2.4. $\bullet$

Now we can give condition for |\mbox{\pi(\delta)}\cap\mbox{T(X)}|>1.

Theorem 2.6

Let $\delta$ be a compatible quasi-proximity, \mbox{{{{\cal V}}}}_{\delta} be transitive and ${{{\cal B}}}$ be the l-base associated with \mbox{{{{\cal V}}}}_{\delta}. In this case there exists \mbox{{{{\cal V}}}}\in\mbox{\pi(\delta)}\cap\mbox{T(\tau)},\ \mbox{{{{\cal V}}}}\not=\mbox{\mbox{${{{\cal V}}}$}_{\delta}} if and only if either

1. there exists a system of sets \{N_{i}:i\in\mbox{\rm I!\bf N}\}\ such\ that\ N_{i}\in\mbox{{{{\cal B}}}},N_{i}\subseteq N_{i+1},N_{i}\not=N_{i+1}\ and\ \bigcup_{1}^{\infty}N_{i}\in\mbox{{{{\cal B}}}} or

2. there exists a system of sets \{N_{i}:i\in\mbox{\rm I!\bf N}\}\ such\ that\ N_{i}\in\mbox{{{{\cal B}}}},N_{i+1}\subseteq N_{i},N_{i}\not=N_{i+1}\ and\ \bigcap_{1}^{\infty}N_{i}\in\mbox{{{{\cal B}}}}.

Proof: Let us prove first the sufficiency and suppose that condition 1 holds. Let \alpha=\{X;N_{i}:i\in\mbox{\rm I!\bf N}\}{{\ \rm and\ }}\mbox{{{{\cal V}}}}=\mbox{\rm fil{}{X\times X}}\{\mbox{\mbox{${{{\cal V}}}$}{\delta}},U_{\alpha}\}. By 2.4 \mbox{{{{\cal V}}}}\in\mbox{\pi(\delta)}\cap\mbox{T(\tau)} since if $A\subseteq X$ then $U_{\alpha}(A)$ is either a finite union of $N_{i}$’s or if it is infinite then it equals to \bigcup_{1}^{\infty}N_{i}\in\mbox{{{{\cal B}}}}. If 2 holds then let \alpha=\{X;\bigcap_{1}^{\infty}N_{i}\}\cup\{N_{i}:i\in\mbox{\rm I!\bf N}\}{{\ \rm and\ }}\mbox{{{{\cal V}}}}=\mbox{\rm fil{}{X\times X}}\{\mbox{\mbox{${{{\cal V}}}$}{\delta}},U_{\alpha}\}. Then $U_{\alpha}(A)$ is always a finite union so it is in ${{{\cal B}}}$. Finally by [11]2.7 \mbox{U_{\alpha}}\notin\mbox{\mbox{${{{\cal V}}}$}_{\delta}} in both cases.

Let us prove the necessity. If \mbox{{{{\cal V}}}}\in\mbox{\pi(\delta)} is transitive, \mbox{{{{\cal V}}}}\not=\mbox{\mbox{${{{\cal V}}}$}_{\delta}} then there is a transitive U\in\mbox{{{{\cal V}}}}{{\ \rm such\ that\ }}\{U(x):x\in X\} is infinite. If we suppose that there is A=\{x_{i}\in X:i\in\mbox{\rm I!\bf N}\}{{\ \rm such\ that\ }} U(x_{n})\not\subseteq\bigcup_{i=1}^{n-1}U(x_{i})\ (\forall n\in\mbox{\rm I!\bf N}) then condition 1 holds since by 2.5 U(x_{i})\in\mbox{{{{\cal B}}}},\ U(A)\in\mbox{{{{\cal B}}}} and the system \{\cup_{j=1}^{n}U(x_{j}):n\in\mbox{\rm I!\bf N}\} will work.

In case there exists no such system then there is a finite set $A_{1}\subseteq X$ such that $X=U(A_{1})$. Since $\{U(x):x\in X\}$ is infinite then there exists $x_{1}\in A_{1}$ such that $\{U(y):y\in U(x_{1})\}$ is infinite by $U$ being transitive. Then there is a finite $A_{2}\subseteq X$ such that $U(A_{2})=U(x_{1})$ and there exists $x_{2}\in A_{2}{{\ \rm such\ that\ }}\{U(y):y\in U(x_{2})\}$ is infinite. Let us continue this process. We get a system \{U(x_{i}):i\in\mbox{\rm I!\bf N}\}{{\ \rm such\ that\ }}U(x_{i})\supset U(x_{i+1}). $U(x_{i})\not=U(x_{i+1})$ can also be assumed. By 2.3 \bigcap_{1}^{\infty}U(x_{i})\in\mbox{{{{\cal B}}}}. $\bullet$

Corollary 2.7

If $\delta_{1}$ is coarser than $\delta_{2}$, \mbox{{{{\cal V}}}}_{\delta_{1}},\mbox{{{{\cal V}}}}_{\delta_{2}} are transitive, |\pi(\delta_{1})\cap\mbox{T(\tau)}|>1 then |\pi(\delta_{2})\cap\mbox{T(\tau)}|>1.

Proof: Obviously \mbox{{{{\cal B}}}}(\delta_{1})\subseteq\mbox{{{{\cal B}}}}(\delta_{2}) and apply 2.6. $\bullet$

We will need definitions and a theorem from [12] dealing with p-filters.

Definition 2.8

([12]2.2) Let N\subseteq\mbox{\rm I!\bf N}. We call a subset $H$ of $N$ a pile of $N$ or an $N$-pile or simply pile (if there is no misunderstanding) if

1. i,j\in H\ and\ i<k<j,\ k\in\mbox{\rm I!\bf N} implies that $k\in H$ and

2. $H$ is maximal for the previous property.

Definition 2.9

([12]2.3) Let again N\subseteq\mbox{\rm I!\bf N}. We say that Z\subseteq\mbox{\rm I!\bf N} is admissible for $N$ if there exists k\in\mbox{\rm I!\bf N}\ such\ that\ H being an $N$-pile implies that $|H\cap Z|\leq k$.

Definition 2.10

([12]2.4) We call a filter $\sigma$ on $\rm I\!\bf N$ a p-filter if, whenever N\in\mbox{\sigma}\ and\ Z is admissible for $N$, then N-Z\in\mbox{\sigma}.

Theorem 2.11

([12]2.7) |\{\mbox{\sigma}:\mbox{\sigma} is a p-filter $\}|=2^{2^{\aleph_{0}}}$.

Now we are ready to prove the main theorem.

Theorem 2.12

Let $\delta$ be a quasi-proximity such that \mbox{{{{\cal V}}}}_{\delta} is transitive. In this case |\mbox{\pi(\delta)}|=1 or |\mbox{\pi(\delta)}|\geq 2^{2^{\aleph_{0}}}, moreover |\mbox{\pi(\delta)}\cap\mbox{T(\tau)}|=1 or $\geq 2^{2^{\aleph_{0}}}$.

Proof: Let ${{{\cal B}}}$ be the l-base associated with \mbox{{{{\cal V}}}}_{\delta}. If |\mbox{\pi(\delta)}|>1 then condition 1 or 2 holds in theorem 2.6.

1. Suppose that there exists a system of sets \{N_{i}:i\in\mbox{\rm I!\bf N}\}{{\ \rm such\ that\ }}N_{i}\in\mbox{{{{\cal B}}}},N_{i}\subseteq N_{i+1},N_{i}\not=N_{i+1}{{\ \rm and\ }}\bigcup_{1}^{\infty}N_{i}\in\mbox{{{{\cal B}}}}.

If A\subseteq\mbox{\rm I!\bf N}{{\ \rm then\ }} let

[TABLE]

Then $\alpha_{A}$ is an interior preserving open cover of $X$. Let $U_{A}(x)=U_{\alpha_{A}}(x)=\cap\{M\in\alpha_{A}:x\in M\}$. If $\sigma$ is a p-filter on $\rm I\!\bf N$ then let

[TABLE]

It is obvious that \mbox{{{{\cal V}}}}_{\sigma}\in\mbox{T(\tau)}. It is also straightforward that $U_{A_{1}}\cap U_{A_{2}}=U_{A_{1}\cap A_{2}}$. By 2.4, \mbox{\mbox{${{{\cal V}}}$}_{\sigma}}\in\mbox{\pi(\delta)}.

We show that if \sigma_{1}\not=\sigma_{2}{{\ \rm then\ }}\mbox{{{{\cal V}}}}_{\sigma_{1}}\not=\mbox{{{{\cal V}}}}_{\sigma_{2}}. This yields immediately that the cardinality of the set of all p-filters on $\rm I\!\bf N$ is less than or equal to the cardinality of \mbox{\pi(\delta)}\cap\mbox{T(\tau)}, hence |\mbox{\pi(\delta)}\cap\mbox{T(\tau)}|\geq 2^{2^{\aleph_{0}}}.

Now let \mbox{{{{\cal V}}}}_{\sigma_{1}}\subseteq\mbox{{{{\cal V}}}}_{\sigma_{2}} then we will show that $\sigma_{1}\subseteq\sigma_{2}$, which is enough to prove since with the opposite case it implies the required statement.

Let $A_{1}\in\sigma_{1}$. Then U_{A_{1}}\in\mbox{{{{\cal V}}}}_{\sigma_{1}} hence  there are A_{2}\in\sigma_{2}{{\ \rm and\ }}P=U_{\beta}\in\mbox{\mbox{${{{\cal V}}}$}_{\delta}}{{\ \rm such\ that\ }}U_{\beta}\cap U_{A_{2}}\subseteq U_{A_{1}} where $\beta$ is a finite subset of ${{{\cal B}}}$. Let |\{P(x):x\in X\}|=k\in\mbox{\rm I!\bf N}. Let $H\subseteq A_{2}$ be an $A_{2}$-pile such that H=\{r\in\mbox{\rm I!\bf N}:p<r<q\}. Suppose that $i\in H-A_{1}$ and let us fix $x_{i}\in N_{i}-N_{i-1}$. Then $U_{A_{1}}(x_{i})=N_{i}{{\ \rm and\ }}U_{A_{2}}(x_{i})=N_{q}$ and we get that $N_{q}\cap P(x_{i})\subseteq N_{i}$ which implies that P(x_{i})\cap(N_{q}-N_{i})=\mbox{\emptyset}. If j\in H-A_{1}{{\ \rm such\ that\ }}i<j{{\ \rm then\ }}x_{j}\in P(x_{j})\cap(N_{q}-N_{i})\not=\mbox{\emptyset} so $P(x_{i})\not=P(x_{j})$ which verifies that the function $f$ defined by $f(i)=P(x_{i})\ (i\in H-A_{1})$ is injective. A similar argument applies if $q=\infty$. Hence $|H-A_{1}|\leq k$ and there is Z=A_{2}-A_{1}\subseteq\mbox{\rm I!\bf N}{{\ \rm such\ that\ }}A_{2}-Z\subseteq A_{1} where $Z$ is admissible for $A_{2}$. Since \mbox{\sigma}_{2} is a p-filter then A_{1}\in\mbox{\sigma}_{2}{{\ \rm and\ }}\mbox{\sigma}_{1}\subseteq\mbox{\sigma}_{2}.

2. Suppose that there exists a system of sets \{N_{i}:i\in\mbox{\rm I!\bf N}\}{{\ \rm such\ that\ }}N_{i}\in\mbox{{{{\cal B}}}},N_{i+1}\subseteq N_{i},N_{i}\not=N_{i+1}{{\ \rm and\ }}\bigcap_{1}^{\infty}N_{i}\in\mbox{{{{\cal B}}}}.

Let $N_{0}=X$. If A\subseteq\mbox{\rm I!\bf N}{{\ \rm then\ }} let

[TABLE]

Then $\alpha_{A}$ is an interior preserving open cover of $X$. Let $U_{A}(x)=\bigcap\{M\in\alpha_{A}:x\in M\}$. If $\sigma$ is a p-filter on $\rm I\!\bf N$ then let

[TABLE]

It is obvious that \mbox{{{{\cal V}}}}_{\sigma}\in\mbox{T(\tau)}. It is also straightforward that $U_{A_{1}}\cap U_{A_{2}}=U_{A_{1}\cap A_{2}}$. By 2.4, \mbox{\mbox{${{{\cal V}}}$}_{\sigma}}\in\mbox{\pi(\delta)}.

We show that if \sigma_{1}\not=\sigma_{2}{{\ \rm then\ }}\mbox{{{{\cal V}}}}_{\sigma_{1}}\not=\mbox{{{{\cal V}}}}_{\sigma_{2}}. This yields immediately that |\mbox{\pi(\delta)}\cap\mbox{T(\tau)}|\geq 2^{2^{\aleph_{0}}}.

Now let \mbox{{{{\cal V}}}}_{\sigma_{1}}\subseteq\mbox{{{{\cal V}}}}_{\sigma_{2}} then we will show that $\sigma_{1}\subseteq\sigma_{2}$, which is enough to be proved.

Let $A_{1}\in\sigma_{1}$. Then U_{A_{1}}\in\mbox{{{{\cal V}}}}_{\sigma_{1}} hence  there are A_{2}\in\sigma_{2}{{\ \rm and\ }}P=U_{\beta}\in\mbox{\mbox{${{{\cal V}}}$}_{\delta}}{{\ \rm such\ that\ }}U_{\beta}\cap U_{A_{2}}\subseteq U_{A_{1}} where $\beta$ is a finite subset of ${{{\cal B}}}$. Let |\{P(x):x\in X\}|=k\in\mbox{\rm I!\bf N}. Let $H\subseteq A_{2}$ be an $A_{2}$-pile such that H=\{r\in\mbox{\rm I!\bf N}:p<r<q\}. (Possibly $p=0$ or $q=\infty$.) Suppose that $i\in H-A_{1}$ and let us fix $x_{i}\in N_{i}-N_{i+1}$. Then $U_{A_{1}}(x_{i})=N_{i}{{\ \rm and\ }}U_{A_{2}}(x_{i})=N_{p}$ and we get that $N_{p}\cap P(x_{i})\subseteq N_{i}$ which implies that P(x_{i})\cap(N_{p}-N_{i})=\mbox{\emptyset}. If j\in H-A_{1}{{\ \rm such\ that\ }}i<j{{\ \rm then\ }}x_{j}\in P(x_{j})\cap(N_{p}-N_{i})\not=\mbox{\emptyset} so $P(x_{i})\not=P(x_{j})$ which verifies that the function $f$ defined by $f(i)=P(x_{i})\ (i\in H-A_{1})$ is injective. Hence $|H-A_{1}|\leq k$ and there is Z\subseteq\mbox{\rm I!\bf N}{{\ \rm such\ that\ }}A_{2}-Z\subseteq A_{1} where $Z$ is admissible for $A_{2}$. Since \mbox{\sigma}_{2} is a p-filter then A_{1}\in\mbox{\sigma}_{2}{{\ \rm and\ }}\mbox{\sigma}_{1}\subseteq\mbox{\sigma}_{2}. $\bullet$

We know from [13]2.4 that in the class of locally compact, T2 spaces if there is a transitive \mbox{{{{\cal V}}}}\in\mbox{\pi(\delta^{0})} then $X$ is 0-dimensional. With the help of 2.6 one can easily prove that |\mbox{\pi(\delta^{0})}\cap\mbox{T(\tau)}|=1 if $X$ is compact or non-Lindelöf, but [13]2.14 and 3.12 yield a more general result, namely |\mbox{\pi(\delta^{0})}|=1 in these cases. So in the context of this paper only the case in which $X$ is non-compact, Lindelöf and 0-dimensional is interesting.

Corollary 2.13

Let $X$ be locally compact, T2, non-compact, Lindelöf and 0-dimensional. If $\delta$ is a compatible quasi-proximity such that \mbox{{{{\cal V}}}}_{\delta} is transitive then |\pi(\delta)\cap\mbox{T(\tau)}|\geq\mbox{2^{2^{\aleph_{0}}}}.

Proof: By 2.7 it is enough to verify the statement for $\delta^{0}$. In this case \mbox{{{{\cal B}}}}=\{compact-open sets\}\cup\{\mbox{\emptyset},X\}, and there is a strictly increasing sequence $(N_{i})$ of compact-open sets, such that $\cup_{1}^{\infty}N_{i}=X$ and 2.6 and 2.12 are applicable. $\bullet$

Finally we present some open problems which seem to be interesting.

Problem 1

What can we say about |\mbox{\pi(\delta)}| if \mbox{{{{\cal V}}}}_{\delta} is not transitive?

Remark : We note that in the meantime this problem have been solved by Künzi in [8].

Problem 2

Let \mbox{{{{\cal V}}}}_{\delta} be transitive. What can be said about |\mbox{\pi(\delta)}\cap(N(X)-T(X))|? Can it occur that there exists no non-transitive quasi-uniformity in $\pi(\delta)$ if $|\pi(\delta)|>1$?

Problem 3

Is there any connection between $|\pi(\delta_{1})|\ and\ |\pi(\delta_{2})|$ if $\delta_{1}$ is coarser than $\delta_{2}$? Is it true that $|\pi(\delta_{1})|\leq|\pi(\delta_{2})|$ in this case?

Problem 4

Is there always a non-transitive quasi-uniformity that is finer than ${{{\cal P}}}$(assuming that $|\pi(\delta^{1})|>1$)? (see also [7]Remark 1)