Tube Lemma
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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 × Y is the product space, a slice in X × Y, is a set of the form {x} × Y for x X A tube in X × Y, is just a basis element, K × Y, in X × Y containing a slice in X × Y Tube Lemma: Let X and Y be topological spaces with Y compact, and consider the product space X × Y. If N is an open set containing a slice in…mehr

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Produktbeschreibung
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 × Y is the product space, a slice in X × Y, is a set of the form {x} × Y for x X A tube in X × Y, is just a basis element, K × Y, in X × Y containing a slice in X × Y Tube Lemma: Let X and Y be topological spaces with Y compact, and consider the product space X × Y. If N is an open set containing a slice in X × Y, then there exists a tube in 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 × Y X is closed.