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## Main Question or Discussion Point

I am trying to show that the space Cone(L(X,x)) is homeomorphic to P(X,x)

where L(X,x) = {loops in X base point x} and

P(X,x) = {paths in X base point x}

I firstly considered (L(X,x) x I) and tried to find a surjective map to P(X,x) that would quotient out right but i couldnt seem to find it. For example i considered

(l,t) -> p where p is the path such that p(1)=l(t) and they agree naturally before

i.e. F((l,t))(s) = l(ts)

I was wondering if anyone could point me roughly the right way or just chip in with their own thoughts

where L(X,x) = {loops in X base point x} and

P(X,x) = {paths in X base point x}

I firstly considered (L(X,x) x I) and tried to find a surjective map to P(X,x) that would quotient out right but i couldnt seem to find it. For example i considered

(l,t) -> p where p is the path such that p(1)=l(t) and they agree naturally before

i.e. F((l,t))(s) = l(ts)

I was wondering if anyone could point me roughly the right way or just chip in with their own thoughts