Proof of Topology: Compact Subsets in Open Sets

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The discussion confirms that 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 such that AxB is contained in UxV, which is also contained in W. This statement, known as the generalized tube lemma, is true for all topological spaces, not just regular spaces. The proof is based on the property 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.

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

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