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 of open covers of a space is a development for if for each and each open set with , there is some such that any open set in containing the point is contained in . A developable space is one that has a development. A Moore space is a regular developable space.

For a collection of of subsets of a space and for , define . An equivalent way of defining a development: A sequence of open covers of a space is a development for if for each , is a local base at . 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 is a Moore space, then .

**Proof**. Since always holds, we only need to show . To this end, we exhibit a base with . Let be a development for . Let be a network with cardinality .

For each , choose open set such that . Let and . Note that . Because is a network, each is a cover of . To see this, let . Choose some such that . There is some such that . Then . For each , . The sequence works like a development. We have just shown that is a base for .

**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.

**Reference**

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

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