x2.1. Topological Spaces Let X be a set. A family T of subsets of X is a topology for X if T has the following three properties: (a) Both X and the empty set belong to T . (b) Any union of sets in T belongs to T . (c) Any finite intersection of sets in T belongs to T . A topological space is a pair (X ; T ), where X is a set and T is a topology for X . The sets in T are called open sets. Example. (a) Let X be a set. Let T0 = fX ; ;g. Then T0 is a topology for X . It is called the trivial topology. It is the smallest topology for X in the sense that if T is a topology for X , then T0 ⊂ T . (b) Let X be a set. Let T1 be the family of all subsets of X . Then T1 is a topology for X . It is called the discrete topology. It is the largest topology for X in the sense that if T is a topology for X , then T ⊂ T1. (c) Let X be a metric space. Then the family of open subsets (defined in terms of the metric) of X is a topology for X . It is called the metric topology. (d) Let Y = f1; 2; 3;::: g. Let Yn = f1;:::; ng for each positive integer n. The family of the sets Yn, together with ; and Y , is a topology for Y . A topological space X is said to be metrizable if the topology for X is the metric topology associated with some metric on X . Example. (a) Let X be a set with more than one element. Then the topological space X with the trivial topology is not metrizable. One way to see this is to note that in a metric space the complement of a singleton needs to be open. (b) Let X be a set. Then the topological space X with the discrete topology is metrizable. It is the metric topology associated with the discrete metric. Let (X ; T ) be a fixed topological space. A subset S of X is said to be closed if X n S is open. 1.0. Theorem. (a) Both ; and X are closed sets. (b) Any intersection of closed sets is closed. (c) A finite union of closed sets is closed. A subset S of X is said to be a neighborhood of a point x if there is an open set U such that x 2 U and U ⊂ S. A point x 2 X is said to be an interior point of S if S is a neighborhood of x. The interior of S is the set of interior points of S. It is denoted by int(S). It is clear that int(S) ⊂ S. 1.1. Theorem. A subset S of a topological space is open iff S = int(S). 1.2. Theorem. If S is a subset of a topological space, then int(S) is open. A point x 2 X is said to be an adherent point of a subset S of X if S meets every neighborhood of x.The closure of S, denoted by S, is the set of adherent points of S. Evidently S ⊂ S. Let S c = X n S. 1.3. Theorem. Let S be a subset of a topological space X . c (a) S = int(S c ): (b) S is closed iff S = S. 1.4. Theorem. If S is a subset of a topological space X , then S is closed. Example. Let Y be the topological space in the first example of this section. Let S = f1g. Then S = Y . A sequence fxj g in a topological space X is said to converge to a point x 2 X if for each neighborhood U of x there is an integer N such that xj 2 U for all j > N. Note that a sequence may converge to more than one point. Example. (a) Let X be a set and let T be the family of subsets U of X such that X n U is finite, together with the empty set. Then T is a topology for X , called the cofinite topology of X . (b) Let X have the cofinite topology. Then a sequence converges to x 2 X iff each point in X other than x appears in the sequence at most finitely many times. (c) Let Z have the cofinite topology. Then the sequence f1; 2; 3;::: g converges to each point of Z. 1.5. Theorem. If S is a subset of a topological space X and if a sequence fxj g in S converges to x 2 X , then x 2 S. The converse of Theorem 1.5 is not true. There may be some x 2 S such that no sequence in S converges to x. Recall that a set is said to be countable if it is empty, or finite, or denumerable (i.e., having the same cardinality as N, the set of positive integers). Example. Let X be an uncountable set and let T be the family of subsets U of X such that X n U is countable, together with the empty set. Then T is a topology for X .A sequence fxj g in X is convergent iff it is eventually constant. I.e., fxj g converges to x iff there is an integer N such that xj = x for all j > N. Let z 2 X and let S = X n fzg. Then S = X . It follows that z 2 S, but no sequence in S converges to z. Let S be a subset of a topological space X . The boundary of S is @S := S \ S c . It is clear that @S = @(S c ). A point x 2 X is said to be a boundary point of S if x 2 @S. The exterior of S is int(S c ). 1.6. Theorem. Let S be a subset of a topological space X . (a) X is the disjoint union of the interior of S, the boundary of S, and the exterior of S: X = int(S) [ @S [ int(S c ). (b) S = int(S) [ @S, S c = @S [ int(S c ). 1.7. Theorem. Let S be a subset of a topological space X . (a) int(S) equals the union of all open sets contained in S. (b) S equals the intersection of all closed sets containing S..