Lecture on Topology Definitions and Examples and Preliminary

Lecture on Topology Definitions and Examples and Preliminary Terms By Mr. Dnyaneshwar R. Nhavi (Assi. Prof. K. C. E. ’s PGCSTR, Jalgaon)

Topological Spaces �Definition – Topology A topology T on a set X is a collection of subsets of X such that 1. The topology T contains both the empty set ∅ and X. 2. Every union of elements of T belongs to T. 3. Every finite intersection of elements of T belongs to T.

Related Terms: topological space (X, T ) consists of a set X and a topology T. �Every metric space (X, d) is a topological space. In fact, one may define a topology to consist of all sets which are open in X. This particular topology is said to be induced by the metric. �The elements of a topology are often called open. �A

Examples of topological spaces �The discrete topology on a set X is defined as the topology which consists of all possible subsets of X. �The indiscrete topology on a set X is defined as the topology which consists of the subsets ∅ and X only. �Every metric space (X, d) has a topology which is induced by its metric. It consists of all subsets of X which are open in X.

Closed Sets �Definition – Closed set Suppose (X, T ) is a topological space and let A ⊂ X. We say that A is closed in X , if its complement X − A is open in X.

Theorem: �If a subset A ⊂ X is closed in X , then every sequence of points of A that converges must converge to a point of A. �Both ∅ and X are closed in X �Finite unions of closed sets are closed. �Arbitrary intersections of closed sets are closed.

Closure of a set �Definition – Closure Suppose (X, T ) is a topological space and let A ⊂ X. The closure of A (Cl(A)) is defined as the smallest closed set that contains A. It is thus the intersection of all closed sets that contain A. Examples: �The interval A = [0, 1) has Cl(A)= [0, 1]. �The interval A = (0, 1) has Cl(A)= [0, 1].

�Theorem: One has A ⊂ Cl(A), for any set A. 2. If A ⊂ B, then Cl(A)⊂ Cl(B)as well. 3. The set A is closed if and only if Cl(A)= A. 4. The closure of A is itself, namely Cl(Cl(A)= Cl(A). 1.

Interior of a set �Definition – Interior Suppose (X, T ) is a topological space and let A ⊂ X. The interior A◦ of A is defined as the largest open set contained in A. It is thus the union of all open sets contained in A. Examples: = [0, 1] has interior A ◦ = (0, 1). �The interval A = [0, 1) has interior A ◦ = (0, 1). �The interval A

�Theorem: 1. A ◦ ⊂ A for any set A. 2. If A ⊂ B, then A ◦ ⊂ B ◦ as well. 3. The set A is open if and only if A ◦ = A. 4. The interior of A ◦ is itself, namely ( A ◦ ) ◦ = A◦.

Limit points �Definition – Limit point Let (X, T ) be a topological space and let A ⊂ X. We say that x is a limit point of A if every neighbourhood of x intersects A at a point other than x. �Theorem: Let (X, T ) be a topological space and let A ⊂ X. If Aj is the set of all limit points of A, then the Cl(A) = A ∪ Aj.

�Properties: 1. limit points of A are limits of sequences of points of A. 2. The set A = {1/n : n ∈ N} has only one limit point, namely x = 0. 3. Every point of A = (0, 1) is a limit point of A, while Aj = [0, 1]. 4. A set is closed if and only if it contains its limit points.

References �Topology by J. R. Mukres. �Introduction to General Topology by K. D. Joshi �Ppts on topology by google. com

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