Homotopy Equivalence
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12-5-2021
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Homotopy Equivalence
Two topological spaces 
and 
are homotopy equivalent if there exist continuous maps 
and 
, such that the composition 
is homotopic to the identity 
on 
, and such that 
is homotopic to 
. Each of the maps 
and 
is called a homotopy equivalence, and 
is said to be a homotopy inverse to 
(and vice versa).
One should think of homotopy equivalent spaces as spaces, which can be deformed continuously into one another.
Certainly any homeomorphism 
is a homotopy equivalence, with homotopy inverse 
, but the converse does not necessarily hold.
Some spaces, such as any ball 
, can be deformed continuously into a point. A space with this property is said to be contractible, the precise definition being that 
is homotopy equivalent to a point. It is a fact that a space 
is contractible, if and only if the identity map 
is null-homotopic, i.e., homotopic to a constant map.
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