Tube lemma

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In mathematics, in the field of topology, the tube lemma is a result which states that if X and Y are topological spaces with Y compact, then in the product space, any open cover of a slice over Y also covers a tube about that slice. More formally, if there is an open cover C of open sets in X \times Y of the set \{x\} \times Y for some x \in X, then there exists a neighborhood U of x such that C also covers U \times Y.

[edit] Proof

Consider the projection C' = \pi_Y C =  \{\{y \in Y | \exists x . (x,y)\in V\}\, |\, V \subseteq X \times Y, V \in C\} of C to an open cover of open sets of Y. Evidently C' actually covers Y. By assumption Y is compact, so there is a finite subcover C'_{sub} \subseteq C'. This subcover must have arisen as the projection of open sets back in the original open cover of X \times Y; that is, there is a subset C_{sub} \subseteq C such that πYCsub = C'sub.

Now Csub is a finite open cover of a slice over Y, and by considering the projection πXCsub of Csub down to X, and taking the (finite!) intersection of all open sets in it, one obtains the open neighborhood U of x that meets the requirements of the lemma.