Timeline for The self-duality of topological compactness
Current License: CC BY-SA 2.5
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| Jun 15, 2020 at 7:27 | history | edited | CommunityBot |
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| Feb 5, 2011 at 0:40 | comment | added | Peter LeFanu Lumsdaine | @Theo, @Qiaochu: Yes, it is very often ambiguous to talk about the dual of a concept, since most concepts have many equivalent formulations, which will typically no longer be equivalent when dualised. (The main exceptions, of course, being concepts primarily defined in explicitly categorical terms.) | |
| Feb 4, 2011 at 22:27 | comment | added | David Roberts♦ | compact and opcompact may mean the same thing for topological spaces, but they may separate when considered for locales. There is a concept called overtness (ncatlab.org/nlab/show/overt+space) which for spaces holds always, but not always for locales (it depends on the classicality of your foundations, or equivalently, if you are working in a topos or not. Classically all locales are overt). Overtness is in a sense a logical dual to compactness, by swapping quantifiers $\forall \leftrightarrow \exists$. | |
| Feb 4, 2011 at 21:56 | comment | added | David Feldman | @Theo Years ago a friend and I wrote an unpublished and I fear flawed paper about maximal filters in the partition lattice of a discrete set. If we'd managed to get that right we would have moved on the the topological situation. One has principal co-ultrafilters - a partition with one doubleton and all other cells singletons. And then non-principal co-ultrafilters seem to have something to do with ultrafilters (one gets "big" partition lattice filters from an ultrafilter $\mu$ by making a set in $\mu$ a cell and any every element in the complement a singleton). But not big enough. | |
| Feb 4, 2011 at 21:08 | comment | added | Theo Johnson-Freyd | @Qiaochu: I'd say that's "a" dual, not "the" dual. But, yes, I agree that that's a very natural dual notion to compactness. Another dual notion might be to develop a theory of "coultrafilters", and decide when they "nverge". | |
| Feb 4, 2011 at 20:21 | comment | added | Qiaochu Yuan | In the language of ultrafilters, a space is compact iff every ultrafilter converges to at least one point. So in some sense the "dual" of compactness is the Hausdorff property: every ultrafilter converges to at most one point. I learned this from Terence Tao's blog. | |
| Feb 4, 2011 at 19:29 | history | asked | David Feldman | CC BY-SA 2.5 |