In mathematics, particularly topology, the tube lemma is a useful tool in order to prove that the finite product of compact spaces is compact. It is in general, a concept of point-set topology. Before giving the lemma, we note the following terminology: * If X and Y a topological spaces and X x Y is the product space, a slice in X x Y, is a set of the form {x} x Y for x ? X * A tube in X x Y, is just a basis element, K x Y, in X x Y containing a slice in X x Y Tube Lemma: Let X and Y be topological spaces with Y compact, and consider the product space X x Y. If N is an open set containing a slice in X x Y, then there exists a tube in X x Y containing this slice and contained in N. Using the concept of closed maps, this can be rephrased concisely as follows: if X is any topological space and Y a compact space, then the projection map X x Y ? X is closed.

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