In this post, it is shown that if $X$ is a locally-compact Hausdorff space then $C(X,Y\times Z)\cong C(X,Y)\times C(X,Z)$ are homeomorphic when equipped with the compact-open topologies. I'm having trouble finding this in a book/ original paper. Does anyone know a reference?
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1Is $X$ Hausdorff too? – Henno Brandsma Mar 27 '20 at 09:53
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Indeed, I made the adjustment. – Mar 27 '20 at 09:55
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1Are you interested in a reference or a proof? I feel the proof should be elementary. As to reference, you could start by investigating the references of these two entries https://www.encyclopediaofmath.org/index.php/Exponential_law_(in_topology) https://ncatlab.org/nlab/show/exponential+law+for+spaces (the exponential law of your equation is $(Y\times Z)^X = Y^X \times Z^X$, but its not listed in these entries) – s.harp Mar 27 '20 at 14:01
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1It is true for arbitrary $X, Y, Z$. In Dugundji, James, "Topology", Allyn and Bacon Inc., Boston, 1966, it occurs as an exercise to Section 5 of Chapter XII. – Paul Frost Mar 27 '20 at 16:52
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Perfect. I'd accept that as an aswer if you would like to post it :) – Mar 27 '20 at 17:21
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I posted my comment as an answer. – Paul Frost Mar 28 '20 at 10:37
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It is true for arbitrary $X,Y,Z$. In
Dugundji, James, "Topology", Allyn and Bacon Inc., Boston, 1966
it occurs as an exercise to Section 5 of Chapter XII.
Paul Frost
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