Proof of Topology: Compact Subsets in Open Sets

[ForMyThunder]

If AxB is a compact subset of XxY contained in an open set W in XxY, then there exist open sets U in X and V in Y with AxB contained in UxV contained in W.

Is this true for all spaces XxY? Or does it hold for only regular spaces?

 

[micromass]

This statement is sometimes known as the generalized tube lemma. It holds in any topological space. Regularity is not required.

 

[Greg Bernhardt]

This statement is true for all spaces XxY, not just regular spaces. The proof relies on the fact that open sets in a product space are generated by sets of the form UxV, where U is open in X and V is open in Y. Therefore, any compact subset AxB of XxY can be contained in a finite union of sets of this form, which can then be contained in a single open set UxV.