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## GENERAL TOPOLOGY AND FUNDAMENTAL GROUPS

by: Mrs. Dedric Little

16

0

1

# GENERAL TOPOLOGY AND FUNDAMENTAL GROUPS MTH 631

Marketplace > Oregon State University > Mathematics (M) > MTH 631 > GENERAL TOPOLOGY AND FUNDAMENTAL GROUPS
Mrs. Dedric Little
OSU
GPA 3.79

D. Garity

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COURSE
PROF.
D. Garity
TYPE
Class Notes
PAGES
1
WORDS
KARMA
25 ?

## Popular in Mathematics (M)

This 1 page Class Notes was uploaded by Mrs. Dedric Little on Monday October 19, 2015. The Class Notes belongs to MTH 631 at Oregon State University taught by D. Garity in Fall. Since its upload, it has received 16 views. For similar materials see /class/224440/mth-631-oregon-state-university in Mathematics (M) at Oregon State University.

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Date Created: 10/19/15
Embedding Theorem Other Major Theorems Def A space is separable if it has a countable dense subset Recall that in a metric space this is equivalent to having a countable basis Embedding Theorem Every separable metric space is homeomorphic to a subspace of the Hilbert Cube E l 0 E 01 0 In fact finite dimensional separable metric spaces are homeomorphic to subspaces of H for some n We won t prove this Urysohn Metrization Theorem Every regular space with a countable basis is metrizable Cantor Image Theorem Every separable metric space is the continuous image of a subspace of the Cantor set with em rFallZOOE Embedding Tneoiem iA Mth esi rFallZOOE Embedding Tneoiem 24 Proof of embedding theorem Step 1 Given separable X d replace d by 3 Step 2 Let U1 U2 be a countable basis for X Define f X gt 01 by I X 3XX U Each 1 is continuous Step 3 Define f X gt I by fX f1 X7f2X7 f is continuous since each 1 is Step 4 f X gt Z E fX is 11 onto and continuous Proof Continued Step 5 Check that f 1 is continuous or equivalently that 1 takes open sets in X to open sets in Z with em rFallZOOE Embedding Tneoiem Mth esi rFallZOOE Embedding Tneoiem

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