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  1.3 The Finite-Closed Topology It is usual to define a topology on a set by stating which sets are open. However, sometimes it is more natural to describe the topology by saying which sets are closed. The next definition provides one such example. 1.3.1 Definition. Let X be any non-empty set. A topology τ on X is called the finite-closed topology or the cofinite topology if the closed subsets of X are X and all finite subsets of X; that is, the open sets are Ø and all subsets of X which have finite complements. Once again it is necessary to check that τ in Definition 1.3.1 is indeed a topology; that is, that it satisfies each of the conditions of Definitions 1.1.1. Note that Definition 1.3.1 does not say that every topology which has X and the finite subsets of X closed is the finite-closed topology. These must be the only closed sets. [Of course, in the discrete topology on any set X, the set X and all finite subsets of X are indeed closed, but so too are all other subsets of X....

1.2 Open Sets, Closed Sets, and Clopen Sets Rather than continually refer to “members of τ ", we find it more convenient to give such sets a name. We call them “open sets”. We shall also name the complements of open sets. They will be called “closed sets”. This nomenclature is not ideal, but derives from the so-called “open intervals” and “closed intervals” on the real number line. We shall have more to say about this in Chapter 2. 1.2.1 Definition. Let (X, τ) be any topological space. Then the members of τ are said to be open sets. 1.2.2 Proposition. If (X, τ) is any topological space, then (i) X and Ø are open sets, (ii) the union of any (finite or infinite) number of open sets is an open set, and (iii) the intersection of any finite number of open sets is an open set. Proof. Clearly (i) and (ii) are trivial consequences of Definition 1.2.1 and Definitions 1.1.1 (i) and (ii). The condition (iii) follows from Definition 1.2.1 and Exercises 1.1 #4. On reading Proposition 1.2.2, a ...

 Chapter 1 Topological Spaces Introduction Tennis, football, baseball and hockey may all be exciting games but to play them you must first learn (some of) the rules of the game. Mathematics is no different. So we begin with the rules for topology. This chapter opens with the definition of a topology and is then devoted to some simple examples: finite topological spaces, discrete spaces, indiscrete spaces, and spaces with the finite-closed topology. Topology, like other branches of pure mathematics such as group theory, is an axiomatic subject. We start with a set of axioms and we use these axioms to prove propositions and theorems. It is extremely important to develop your skill at writing proofs. Why are proofs so important? Suppose our task were to construct a building. We would start with the foundations. In our case these are the axioms or definitions – everything else is built upon them. Each theorem or proposition represents a new level of knowledge and must be firmly anchore...