week26 - Topology (Part 2) Original Notes adopted from...

Info iconThis preview shows pages 1–2. Sign up to view the full content.

View Full Document Right Arrow Icon
Topology (Part 2) Original Notes adopted from April 9, 2002 (Week 26) © P. Rosenthal , MAT246Y1, University of Toronto, Department of Mathematics typed by A. Ku Ong Let X be a topological space. Let S X The relative topology on S (as a subset of X) is the topology where a subset U of S is open if there exists V, an open subset of X such that U = V S Eg. X = R usual topology S = [0,1]. 0 ¾ 1 7 U = (3/4, 1]. (is an open subset on relative topology but not open in R) Recall: If f: X Y is continuous, if f -1 (U) is open (in X) whenever U is open (in Y). Definition: The topological space X is disconnected if there exists U,V open subsets of X such that X = U U V & U V = (and neither U nor V is ), If there is no such disconnection, then X is connected. Theorem: If X is connected topological space, and f:X topological space in the relative topology as a subset of Y Proof: Suppose f(x) was disconnected. Then f(x) = U, Union U 2 , each U i is open subset of f(x) U 1 U 2 = , U 1 0, U 2 0.
Background image of page 1

Info iconThis preview has intentionally blurred sections. Sign up to view the full version.

View Full DocumentRight Arrow Icon
Image of page 2
This is the end of the preview. Sign up to access the rest of the document.

Page1 / 3

week26 - Topology (Part 2) Original Notes adopted from...

This preview shows document pages 1 - 2. Sign up to view the full document.

View Full Document Right Arrow Icon
Ask a homework question - tutors are online