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Oct 16, 2019 at 20:37 comment added Ángel Valencia I wish I thought the notion of "compactification" can be seen as a "nice" functor from a category (for example, all locally compact spaces) to a subcategory of "compact" objects (for example, all compact spaces), but I don't know if it works well in a general case...
Oct 15, 2019 at 21:33 comment added André Henriques Dual question: is there a general theory of "making things discrete" [think: find an epimorphism from a set]? Examples: underlying set of a manifold; set of $k$-rational points of a variety; underlying vector space of a Banach space... Once again, if you want the result of this operation to not be too huge, you'll need to make (arbitrary) choices (e.g. the choice of a dense countable set inside a topological space).
Oct 15, 2019 at 12:53 comment added Tim Campion @TobyBartels Thanks, hemicompact spaces are indeed much more general than manifolds! I'm a little confused by the nlab description for arbitrary topological spaces, which I think you wrote judging by the revision history. I've dropped a note at the nforum.
Oct 15, 2019 at 11:54 comment added Toby Bartels The end compactification is for a lot more than just manifolds. It's a point-set concept that works for any hemicompact space (and really any space at all, but as with the Stone–Čech compactification, I'm not sure if it's still a compactification outside of its original context).
Oct 15, 2019 at 8:46 comment added Sebastian Goette Some compacitifactions try to keep some of the geometry of the space they compactify. For example, a conformal compactification tries to keep the conformal structure on the interior. In nice examples, it produces a manifold with boundary or with corners. The boundary at infinity of a Cartan-Hadamard space can be defined so that it keeps track of directions of geodesics. These are examples of the very last bullet in the list.
Oct 15, 2019 at 1:18 comment added Robert Furber This has come up before, but the Bohr compactification is not a compactification in one commonly used sense, because the universal map from an abelian group to its Bohr compactification is not a topological embedding (i.e. the original topology does not agree with the subspace topology on the image). Similarly, the restriction of the Stone-Čech compactification to completely regular spaces is not a necessary part of the construction, but arises because doing so makes the universal mapping an embedding (a space is completely regular iff it is embeddable in a compact Hausdorff space).
Oct 14, 2019 at 22:54 history became hot network question
Oct 14, 2019 at 20:51 history edited Tim Campion CC BY-SA 4.0
added 61 characters in body
Oct 14, 2019 at 17:00 comment added user108998 I think in more geom contexts (ie not point set topology) a fairly neat way to "define" the boundary at infinity would be as a pro-object in the appropriate category, ie the formal diagram lim (X\K) over compact subspaces K. (eg the hemicompact definition I just learned from one of ur links is a srequirement on the indexing category for this diagram). Presumably this says something about compactifications. I think I read this in a recent paper of toen and pantev
Oct 14, 2019 at 15:33 answer added Andreas Blass timeline score: 27
Oct 14, 2019 at 15:15 review Close votes
Oct 16, 2019 at 6:14
Oct 14, 2019 at 15:00 history edited Martin Sleziak
I think that the (compactifications) tag might be suitable for this question
Oct 14, 2019 at 14:48 history asked Tim Campion CC BY-SA 4.0