Homotopy Equivalence

Definition:

Given topological spaces $\QTR{Large}{S}$ and $\QTR{Large}{T}$ we say that $\QTR{Large}{S}$ and $\QTR{Large}{T}$ are homotopy equivalent if there exists continuous maps and , such that ,

and

Note:

Homotopy equivalence is an equivalence relation.
Spaces may be homotopy equivalent without being homeomorphic

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Examples:

Example 1 $\QTR{Large}{[0,1]}$ is homotopy equivalent to $\QTR{Large}{(0,1)}$ . Let be the inclusion map and be the constant map

We need to find such that and Define Exactly the same formula works for

Example 2: $\QTR{Large}{[0,1]}$ is not homotopy equivalent to

Homework Due April 27 : Prove This.

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Proof:

Suppose there were maps and

such that and

since any map is homotopic to by the homotopy where

one cannot expect to find a contradiction in this direction. However, since $\QTR{Large}{[0,1]}$ is connected we know for all $\QTR{Large}{t,}$ or thus or

for Suppose and where is the homotopy. Consider

is a map such that and , a contradiction.

Example 3:

The First Lecture