Hausdorff space

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  If a continuous function is invertable, and maps a compact space onto a hausdorff space, the map is bicontinuous, and implements a homeomorphism. The two spaces are topologically equivalent.

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  this theorem also holds for any locally compact hausdorff space. to review, a space is hausdorff if two points can be separated in disjoint open sets, and a space is locally compact if each point is contained in an open set whose closure is compact.

• Urysohn's lemma

  Proof: If a Hausdorff space X is normal, then it satisfies T4 axiom. According the to Hausdorffness, X is a T2 space and thus it must also be a T1 space. In T1 space, a set with a single point is a closed set.

• Co-Finite Topology

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• De Morgan's Laws

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• subset of S is compact

  Show that S with this topology is compact, but S is not a Hausdorff space. Show that each subset of S is compact and that therefore there are compact subsets of S that are not closed.