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.

Keywords

• topology, compactness, Hausdorff space, limit point