# week26 - Topology(Part 2 Original Notes adopted from April...

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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.

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

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