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A sigma-cellular base metrizes a normal Moore space

Statement

Assume $Z F C$ . Let $X$ be a normal Moore space carrying a base of the form $⋃_{n \in N} B_{n}$ (Basis and subbasis for a topology, and the topology generated by a family of sets) in which every $B_{n}$ is a pairwise disjoint family of open sets. Then $X$ is metrizable.

The normality and development hypotheses are essential to this conclusion: a $T_{1}$ space with a sigma-disjoint base need not be metrizable.

Facts & Assumptions

Given: A normal Moore space $X$ , a development $(G_{i})_{i \in N}$ , and a base $⋃_{n} B_{n}$ whose levels are pairwise disjoint open families.

[F1]

A Moore space is regular and $T_{1}$ and has a development: every $G_{i}$ covers $X$ , and for every open $O ∋ x$ some $i$ satisfies $St (x, G_{i}) \subseteq O$ (Moore spaces and developments).

[F2]

For every open $O ∋ x$ , some member of the displayed base contains $x$ and is contained in $O$ (Basis and subbasis for a topology, and the topology generated by a family of sets).

[F3]

If $A$ is closed, $U$ is open, $A \subseteq U$ , and $X$ is normal, then there is open $D$ with $A \subseteq D \subseteq \overline{D} \subseteq U$ (A space is normal if and only if every closed $A$ inside an open $U$ admits an open $V$ with $A \subseteq V \subseteq \overline{V} \subseteq U$ , Normal spaces and $T_{4}$ spaces, with the source disagreement over whether normality includes $T_{1}$ stated explicitly).

[F4]

A family is discrete when every point has a neighbourhood meeting at most one member; a sigma-discrete open basis of a regular $T_{1}$ space yields a compatible metric in $Z F C$ (Discrete families and $σ$ -locally-finite and $σ$ -discrete bases, Under choice, a space is metrizable if and only if it is regular, $T_{1}$ , and has a $σ$ -discrete basis, The Axiom of Choice).

Proof

technique · direct

1.1givenF1

For $n, i \in N$ , put $B_{n}^{*} = ⋃ B_{n}$ , $W_{n} = X ∖ B_{n}^{*}$ , and $X_{n, i} = X ∖ ⋃ {G \in G_{i} : G \cap W_{n} \neq \emptyset}$ . The set $W_{n}$ is closed and $X_{n, i}$ is closed. Also $X_{n, i} \subseteq B_{n}^{*}$ : a member of the cover $G_{i}$ containing a point of $X_{n, i}$ is disjoint from $W_{n}$ .

2.1F1step 1.1

Every point of $B_{n}^{*}$ belongs to some $X_{n, i}$ . Indeed, if $x \in B \in B_{n}$ , choose $i$ with $St (x, G_{i}) \subseteq B$ by [F1]. Every member of $G_{i}$ containing $x$ is then disjoint from $W_{n}$ , which is precisely $x \in X_{n, i}$ . Empty levels cause no exception: then $B_{n}^{*} = X_{n, i} = \emptyset$ .

2.2F3step 1.1

Apply [F3] to the closed set $X_{n, i}$ inside the open set $B_{n}^{*}$ and obtain open $D_{n, i}$ with $X_{n, i} \subseteq D_{n, i} \subseteq \overline{D_{n, i}} \subseteq B_{n}^{*}$ .

3.1givenF4step 2.2

The family $H_{n, i} = {D_{n, i} \cap B : B \in B_{n}}$ is a discrete family of open sets. A point outside $\overline{D_{n, i}}$ has an open neighbourhood missing every member. A point of $\overline{D_{n, i}}$ lies in $B_{n}^{*}$ by step 2.2 and hence in a unique $B_{0} \in B_{n}$ ; the open neighbourhood $B_{0}$ meets no $D_{n, i} \cap B$ with $B \neq B_{0}$ .

4.1F2F4step 2.1step 2.2step 3.1

The countable union $⋃_{n, i} H_{n, i}$ is an open sigma-discrete basis. To verify the basis property, let $O$ be open and $x \in O$ . By [F2] choose $n$ and $B \in B_{n}$ with $x \in B \subseteq O$ . Step 2.1 gives $i$ with $x \in X_{n, i} \subseteq D_{n, i}$ , so $x \in D_{n, i} \cap B \subseteq O$ and this set belongs to $H_{n, i}$ .

5.1F1F4step 4.1∎

By [F1], $X$ is regular and $T_{1}$ . The sigma-discrete basis of step 4.1 therefore satisfies the reverse direction of Bing's metrization theorem [F4], so $X$ admits a compatible metric.

Remarks

Why the naive block metric fails. Although $B_{n}^{*}$ is open, its complement $W_{n}$ need not be open. Thus $B_{n} \cup {W_{n}}$ need not be an open partition, and agreement on those blocks does not directly define the original topology. Normality and the development are exactly what replace each cellular level by the countable family of discrete open families in step 3.1.
Source route. Step 3.1 is Bing's normal-development conversion from screenable to strongly screenable (Theorem 8). Step 5.1 uses the sigma-discrete-basis form of his metrization theorem (Theorem 3).

Depends on

Used by

Normal screenable Moore spaces are metrizable Theorem

Dependency tree · two levels

23 results within two dependency steps of this one, each drawn at its shortest distance from it. An arrow runs from a result to what uses it, so the chart reads left to right and ends at this result, which carries a heavier outline. Every node is a link to that result. Click elsewhere on the chart to enlarge it.

Sources

R. H. Bing, Metrization of topological spaces (standard reference, not scraped)