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| 13 Jun 2024 |
 gm_z | it is curious that if X is compact we have a homeomorphism X \to \operatorname{mSpec} \mathscr{C}(X,\mathbb{R}) | 02:33:34 |
| gm_z | the proof i know of uses the fact that \mathbb{R} is an ordered ring, but this isn't quite satisfying | 02:34:10 |
| gm_z | (i equip \operatorname{mSpec}R with the zariski subspace topology) | 02:35:09 |
| 17 Jun 2024 |
| gm_z | here is my motivating question: is it true that for every space X, there is an hausdorff other Y, such that X is a quotient of Y? | 03:25:59 |
| gm_z | the answer is yes, through a construction which i proceed to describe. let W be a hausdorff space that admits a decomposition into two dense subsets W = W_0 \sqcup W_+. consider the subspace
\{(x, f) \in X \times W^{|X|} \} where f(z) \in W_0 if z = x and f(z) \in W_+ otherwise (i use W^{|X|} to denote product topology). then this space is hausdorff, and the map (x, f) \mapsto x is the quotient map | 03:36:40 |
| gm_z | now for my actual curiosity | 03:38:40 |
| gm_z | this construction looks suspiciously like the espace étalé for sections | 03:40:06 |
| gm_z | but the espace étalé isn't always hausdorff | 03:40:26 |
| gm_z | what would be the way to rewrite this construction in geometric language? | 03:41:30 |
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