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I was reading Lages Lima's book about the Fundamental Group and to link the natural geometric sense of a homotopy we need to show that given two topological spaces $X$ and $Y$, it is possible to have a one-to-one correspondence between $C(X\times [0,1],Y)$ and $C(X,C([0,1],Y))$. The to-and-fro function back they are almost immediate, since we can $\varphi(H(x,t))=H_t(x)$. The problem is verifying the correct definition of $\varphi$ and $\varphi^{-1}$. I actually have several paths that I have tried without success:

  1. Using the compact-open topology
  2. Not using the compact-open topology, but assuming that $X$ is compact and $Y$ is a metric space.

En lages says that if $X$ is a locally compact Hausdorff space then the correspondence is well defined. Can this be stretched? or must we necessarily have at least that condition for $X$?

Can someone give me an idea please?

Zaragosa
  • 1,679
  • I would say that if a space of function is open in $C(X, C([0, 1], Y)$ in the compact-open topology. then it is open in $C(X\times [0,1], Y)$ -- that seems to be the easy part. For the other one, this may be related? https://math.stackexchange.com/questions/1646205/compact-sets-in-the-product-of-topological-spaces Note also, that the title is a bit different then the quesiton in the text... – Peter Franek Aug 20 '22 at 19:43
  • This reminds me of a general result of category theory – FShrike Aug 20 '22 at 21:15
  • The title of your question differs from its text. I guess the title is correct. – Paul Frost Aug 20 '22 at 22:42
  • @FShrike This is the one variation of the tensor-hom adjunction but it isn‘t purely categorical, as the category of all topological spaces don‘t admit this property. That‘s why topologists work in a so-called convenient category of spaces and e.g. often use compactly generated weak Hausdorff spaces. – Qi Zhu Aug 20 '22 at 22:44

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