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The inspiration of the definition of “compactness” comes from the real number system.
Closed and bounded sets in the real line were considered as an excellent model to show a
generalized version of the compactness in a topological space. Since boundedness is an elusive
concept in general topo-logical space, then the compact properties are analysed to look at some
properties of sets that do not use boundedness. Some of the classical results of this nature are
Bolzano -Weierstrass theorem, whe-re every infinite subset of [a,b] has accumulation point and
Heine-Borel theorem, where every closed and bounded interval [a,b] is compact. Each of these
properties and some others are used to define a generalized version of compactness. Hausdorff
space has compact properties if every compact subset in Hausdorff space is closed and every
infinite Hausdorff space has infinite sequence of non empty and disjoint open sets. Because the
compact properties in the Hausdorff space are satisfied many the-orems in real line could be
expanded. Therefore, these theorems ccould be used in Hausdorff space.


topology, compactness, Hausdorff space, limit point

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