topology definitions

INTRODUCTION TO TOPOLOGY

1. TOPOLOGY MATHEMATICS 1 V.RAMYA

2. TOPOLOGY Topology is the part of geometry that does not depend on specific measurement of distances and angles Topology is the branch of mathematics concerned with basic properties of geometric figures that remain unchanged when they are stretched, shrunk, deformed, or distorted as long as they are not ripped or punctured 2 V.RAMYA

3. PRODUCT TOPOLOGY Let X and Y be topological spaces. The product topology on X × Y is a topology having as basis the collection B of all sets of the form U × V where U is open in X and V is open is Y 3 V.RAMYA

4. PROJECTION MAPPING Let π 1 : X × Y X be defined by π 1 (x,y)=x and Let π 2 : X × Y Y be defined by π 2 (x,y)=y . The mappings π 1 and π 2 are called the projections of X × Y onto its first and second factors respectively. 4 V.RAMYA

5. SUBSPACE TOPOLOGY Let X be a topological space with topology τ . if Y is a subset of X , the collection τ y= {y ∩ u/u ∈ τ } is a topology on Y , called the subspace topology. Then, Y is called a subspace of X and its open sets consists of all intersection of open sets of X with Y . 5 V.RAMYA

6. SUBSPACE TOPOLOGY 6 V.RAMYA

7. INTERIOR Let X be a topological space with topology τ .Let A be a subset of X Interior of A is defined as the union of all open sets contained in A and it’s denoted by int A . 7 V.RAMYA

8. CLOSURE Let X be a topological space with topology τ .Let A be a subset of X Closure of A is defined as the intersection of all closed sets containing A and it’s denoted by Cl A (or) Ᾱ 8 V.RAMYA

9. INTERIOR CLOSURE Interior of A is the largest open set in contained A. Closure of A is the smallest closed set contained in A. Int ( A) (or) Å Cl (A ) (or) Ᾱ Å A A Ᾱ A= open iff A= Å A= closed iff A= Ᾱ V.RAMYA 9 INTERIOR VS CLOSURE

10. convex Let Y be a subset of and ordered set X . then Y is CONVEX in X if for each pair of points a<b of Y . The entire interval (a,b) of points of X lies in Y . 10 V.RAMYA

11. CLOSED SET A subset A of a topological space X is said to be CLOSED , if the set ( X - A ) is open 11 V.RAMYA A

12. LIMIT POINT If A is a subset of a topological space X and if x is a point of X , then x is a limit point of A .It every neighborhood of x intersects a in some point other than x itself. 12 V.RAMYA a b (a) The point x lies in A (b) The point x does not lies in A

13. HAUSDrOFF SPACE A Topological space X is called a hausdorff space if for each pair x 1 , x 2 of distinct points of X , there exists nighbourhoods U 1 , U 2 of x 1 , x 2 respectively , that are disjoint. 13 V.RAMYA

14. 14 V.RAMYA By: