### SIFAT KOMPAK DALAM RUANG HAUSDORFF

#### Abstract

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.

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.

#### Keywords

topology, compactness, Hausdorff space, limit point

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