Moore Space (topology) - Examples and Properties

Examples and Properties

1. Every metrizable space, X, is a Moore space. If {A(n)_x} is the open cover of X (indexed by x in X) by all balls of radius 1/n, then the collection of all such open covers as n varies over the positive integers is a development of X. Since all metrizable spaces are normal, all metric spaces are Moore spaces.
2. Moore spaces are a lot like regular spaces and different to normal spaces in the sense that every subspace of a Moore space is also a Moore space.
3. The image of a Moore space under an injective, continuous open map is always a Moore space. Note also that the image of a regular space under an injective, continuous open map is always regular.
4. Both examples 2 and 3 suggest that Moore spaces are a lot similar to regular spaces.
5. Neither the Sorgenfrey line nor the Sorgenfrey plane are Moore spaces because they are normal and not second countable.
6. The Moore plane (also known as the Niemytski space) is an example of a non-metrizable Moore space.
7. Every metacompact, separable, normal Moore space is metrizable. This theorem is known as Traylor’s theorem.
8. Every locally compact, locally connected space, normal Moore space is metrizable. This theorem was proved by Reed and Zenor.
9. If, then every separable normal Moore space is metrizable. This theorem is known as Jones’ theorem.