SIFAT KOMPAK DALAM RUANG HAUSDORFF - LUH PUTU IDA HARINI (Mathematics Department, Faculty of Mathematics and Natural Sciences, Udayana University, 2007)
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 ageneralized version of the compactness in a topological space. Since boundedness is an elusiveconcept in general topo-logical space, then the compact properties are analysed to look at someproperties of sets that do not use boundedness. Some of the classical results of this nature areBolzano -Weierstrass theorem, whe-re every infinite subset of [a,b] has accumulation point andHeine-Borel theorem, where every closed and bounded interval [a,b] is compact. Each of theseproperties and some others are used to define a generalized version of compactness. Hausdorffspace has compact properties if every compact subset in Hausdorff space is closed and everyinfinite Hausdorff space has infinite sequence of non empty and disjoint open sets. Because thecompact properties in the Hausdorff space are satisfied many the-orems in real line could beexpanded. Therefore, these theorems ccould be used in Hausdorff space.