TOPOLOGY MATHEMATICS 1V.RAMYA

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 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 2V.RAMYA

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 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 3V.RAMYA

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. 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. 4V.RAMYA

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. 5V.RAMYA

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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. 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. 7V.RAMYA

 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  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 8V.RAMYA

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) ÅCl (A) (or) Ᾱ Å A A Ᾱ A= open iff A= Å A= closed iff A= Ᾱ V.RAMYA9

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. 10V.RAMYA

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

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. 12V.RAMYA a b (a) The point x lies in A (b) The point x does not lies in A

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. 13V.RAMYA

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