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Cardinal Numbers, Cardinal Functions, and Pol-Šapirovskii Technique

出版	Central Connecticut State University, 2016
URL	http://books.google.com.hk/books?id=ybKZAQAACAAJ&hl=&source=gbs_api

註釋General Topology is a branch of mathematics that studies fundamental notions such as continuity, compactness and connectedness which are needed and used by most mathematicians. The concept of a topological space plays a central role in this more than 100 years old area of mathematics. A fundamental question in General Topology is when two topological spaces are homeomorphic, or equivalent, in the sense that they are not topologically distinguishable. Cardinal numbers and cardinal functions are often used to answer this question or to help gain additional insight into the characteristics of the topological spaces. For example, to each topological space one could assign cardinal numbers that represent dierent characteristics of the space like cardinality of the entire space, the cardinality of the smallest dense set it contains, etc. Since these cardinal characteristics do not change under homeomorphisms, they are called cardinal invariants. One can also think about each such cardinal invariant as a function that assigns to each topological space a cardinal number (the cardinality of the space, for example). That is why they are sometimes called cardinal functions. In this thesis the reader will and some basic facts from Set Theory and General Topology, along with some historical remarks. These facts are included into the thesis in order to help the reader understand and appreciate some fascinating cardinal inequalities for topological spaces expressed in terms of different cardinal functions and also to learn about their methods of proof, including the so called Pol-Sapirovskii Technique.