Topolojik uzaylarda kompaktlık kavramı
Başlık çevirisi mevcut değil.
- Tez No: 56349
- Danışmanlar: PROF. DR. ASUMAN ILGAZ
- Tez Türü: Yüksek Lisans
- Konular: Matematik, Mathematics
- Anahtar Kelimeler: Topolojik uzaylar, Topological spaces
- Yıl: 1996
- Dil: Türkçe
- Üniversite: Marmara Üniversitesi
- Enstitü: Fen Bilimleri Enstitüsü
- Ana Bilim Dalı: Matematik Ana Bilim Dalı
- Bilim Dalı: Belirtilmemiş.
- Sayfa Sayısı: Belirtilmemiş.
Özet
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Özet (Çeviri)
In this work, we deal with the notion of compactness in topological spaces. Particular attention is given to the basic properties and theorems of the related subject. However, we have stated earlier, namely in the first section, the important properties of the system of real numbers, cartesian spaces, metric spaces and topological spaces. A topological property called compactness is developed for general topological space. For subsets of the space (R, U) compactness turns out to be equivalent to the property of being“closed”and“bounded”. Since these concepts can not be used to define compactness in a general topological space, we need the concept of“open covering”of a topological space. This concept is introduced in the last part of the first section. Indeed compactness is a topological analog of finiteness. That means instead of considering of all points of a space, we consider only finitely many points to study the entire space. That's why many theorems in calculus involve closed and bounded intervals in R, which are all compact spaces. Since the earlier definitions of compactness are related to sequentially compactness and countably compactness, we also deal with them. Furthermore, in metric spaces, we proved that compactness <=> sequentiall compactness <=> countable compactness. In the latter section we concern compact spaces in general; equivalent definitions of them, their separation properties, continuous functions on them and their subspaces. Finally we work with the product topology of compactness. m
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