Hi, everyone:(adsbygoogle = window.adsbygoogle || []).push({});

I am confused about the result that every map from a contractible space X into

any topological space Y is contractible.

I think the caveat here is that the homotopy between any f:X-->Y and c:X-->Y

with c(X)={pt.} is that the homotopy is free, i.e., the endpoints are not fixed.

Still, I find the example of f:I-->^{S^1}, with I the unit interval,

particularly confusing:

Consider the path f(t)=^{e^2iPit}. According to this result, since

I is contractible ( I am contractible?:smile) :, f(t) is trivial, i.e., f(t) is homotopic

to a point.

*But* since^{e^2iPit}does a full loop around^{S^1}, i.e,

f(I)= (^{S^1}), we must tear f(I) open , to be able to homotope it

to a point. Isn't this tearing necessarily discontinuous, even if we do not fix the

endpoints in^{S^1}.?.

If not, what would be an actual homotopy between^{e^2iPit}and {.pt.}?

Thanks.

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# Apparent Contradiction: Every Map from a Contractible Space to any X is trivi

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