CHAPTER III - VARIOUS TOPOLOGICAL SPACES

Abstract

This chapter discusses various topological spaces. A T1-space is characterized as a topological space in which every point forms a closed set. The chapter discusses the proposition that characterizes T2-spaces. A topological space R is a T2-space if every filter converges to at most one point. Every topological space with the weakest topology is not a T1-space if it contains at least two points. Tychonoff's imbedding theorem states that a topological space R is completely regular if it is homeomorphic with a subspace of the product space of the copies of the unit segment. A property that distinguishes normal and fully normal spaces from spaces is that a subspace of a normal space is not necessarily normal. Furthermore, the product space of two fully normal spaces is not necessarily normal. The chapter also discusses Urysohn's lemma. A topological space is compact only if it satisfies the conditions (1) every closed collection with finite intersection property has a non-empty intersection, (2) every filter of R has a cluster point, and (3) every maximal filter of R converges.