A Separation Theorem
Our mathematical travels take us next to a quite different part of topology. It is called general topology or point-set topology and it seeks to discover properties of topological spaces that hold for very broad classes of spaces. A substantial part of the subject concerns spaces that are not necessarily metric and this will be the case of the theorem we will discuss here; it holds for any compact topological space. When we come to apply the theorem, it will be in the setting of metric spaces, because all the spaces that will arise in the analytic context are metric. However, a proof that made use of a metric would be unnecessarily complicated; since the property has nothing to do with distance, there is no point in bringing it up.
KeywordsClosed Subset Compact Space Finite Subset General Topology Separation Property
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