Talk:Exponential object
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[edit] Is Y or Z required to be locally compact Hausdorff?
The text currently seems to confuse Y and Z. Can someone with the knowledge fix it? It looks like a misprint, but I don't want to correct it without being 100% sure.--345Kai 19:22, 2 March 2007 (UTC)
- No, its stated correctly, Y must be locally compact. In particular, if Y is locally compact, then the evaluation map
is continuous. See compact-open topology. -- Fropuff 20:05, 2 March 2007 (UTC)
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- Maybe I should have been more specific where I fear the misprit is. I've highlited it in the corresponding paragraph:
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- In the category of topological spaces, the exponential object ZY exists provided that Y is a locally compact Hausdorff space. In that case, the space ZY is the set of all continuous functions from Y to Z together with the compact-open topology. The evaluation map is the same as in the category of sets. If Y is not locally compact Hausdorff, the exponential object may not exist (the space ZY exists, but fails to be an exponential object because the adjunction with the product only holds when
is locally compact Hausdorff). For this reason the category of topological spaces fails to be cartesian closed.
- In the category of topological spaces, the exponential object ZY exists provided that Y is a locally compact Hausdorff space. In that case, the space ZY is the set of all continuous functions from Y to Z together with the compact-open topology. The evaluation map is the same as in the category of sets. If Y is not locally compact Hausdorff, the exponential object may not exist (the space ZY exists, but fails to be an exponential object because the adjunction with the product only holds when
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- Should the "big Z" be a Y? Thanks for the help! --345Kai 15:06, 4 March 2007 (UTC)
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- Oh right, thanks. I've fixed it now. -- Fropuff 16:32, 4 March 2007 (UTC)
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