Network Weight of Topological Spaces – II

This is a continuation of the discussion on network. In the previous post, I showed that the network weight (the minimum cardinality of a network) coincides with the weight for both metrizable spaces and locally compact spaces. In another post, I showed that this is true for compact spaces. I now show that this is also true for the class of Moore spaces. First, some definitions. A sequence \lbrace{\mathcal{D}_n}\rbrace_{n<\omega} of open covers of a space X is a development for X if for each x \in X and each open set U \subset X with x \in U, there is some n such that any open set in \mathcal{D}_n containing the point x is contained in U. A developable space is one that has a development. A Moore space is a regular developable space.

For a collection of \mathcal{G} of subsets of a space X and for x \in X, define st(x,\mathcal{G})=\bigcup\lbrace{U \in \mathcal{G}:x \in U}\rbrace. An equivalent way of defining a development: A sequence \lbrace{\mathcal{D}_n}\rbrace_{n<\omega} of open covers of a space X is a development for X if for each x \in X, \lbrace{st(x,\mathcal{G}_n):n \in \omega}\rbrace is a local base at x. For a basic introduction to Moore space and the Moore space conjecture, there are numerous places to look in the literature ([1] being one of them).

Theorem. If X is a Moore space, then nw(X)=w(X).

Proof. Since nw(X) \leq w(X) always holds, we only need to show w(X) \leq nw(X). To this end, we exhibit a base \mathcal{B} with \vert \mathcal{B} \lvert \leq nw(X). Let \lbrace{\mathcal{D}_n}\rbrace_{n<\omega} be a development for X. Let \mathcal{N} be a network with cardinality nw(X).

For each N \in \mathcal{N}, choose open set O(n,N) \in \mathcal{D}_n such that N \subset O(n,N). Let \mathcal{B}_n=\lbrace{O(n,N):N \in \mathcal{N}}\rbrace and \mathcal{B}=\bigcup_{n<\omega}\mathcal{B}_n. Note that \lvert \mathcal{B} \lvert \leq nw(X). Because \mathcal{N} is a network, each \mathcal{B}_n is a cover of X. To see this, let x \in X. Choose some V \in \mathcal{D}_n such that x \in V. There is some N \in \mathcal{N} such that x \in N \subset V. Then x \in O(n,N). For each n, \mathcal{B}_n \subset \mathcal{D}_n. The sequence \lbrace{\mathcal{B}_n}\rbrace works like a development. We have just shown that \mathcal{B} is a base for X.

Corollary. The example of Butterfly space is not a Moore space.

The example of the Butterfly (or Bow-tie) space is defined in this previous post. This space has a countable network and the weight of this space is continuum. Thus this space cannot be a Moore space.

[1] Steen, L. A. & Seebach, J. A. [1995] Counterexamples in Topology, Dover Books.


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