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Nullhomotopic Mappings and Contractible Spaces

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I am having problems proving this fact. A space X is contractible if and only if
every map f:X to Y (Y is arbitrary) is nullhomotopic. Similarly show X is contractible
iff ever map f:Y to X is nullhomotopic.

In the first case if Y=X then see that the identity map on X is nullhomotopic. But Im not
sure how to proceed for the rest.

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Solution Summary

Nullhomotopic Mappings and Contractible Spaces are investigated.

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Notation: We let c_x:X ->{x} denote the constant map to {x}.
"o" is composition, "~" is "homotopic to", $1_X$ identity on X

Suppose that X is ...

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