Timeline for Topological characterization of injective metric spaces
Current License: CC BY-SA 3.0
24 events
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| Jun 15, 2020 at 7:27 | history | edited | CommunityBot |
Commonmark migration
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| May 10, 2016 at 4:05 | comment | added | Włodzimierz Holsztyński | (Right, infinite dimensional). | |
| May 10, 2016 at 3:34 | comment | added | Mikhail Ostrovskii | You are right (but in both statements you need to add "infinite-dimensional"), I actually asked whether one can leave the class of Banach spaces (keeping the homeomorphic type of $\ell_2$) and get something injective. | |
| May 10, 2016 at 3:11 | comment | added | Włodzimierz Holsztyński | @MikhailOstrovskii, no separable Banach space is metrically injective. Thus it'd be difficult to show that a separable injective metric space is homemorphic to a Banach space. (All separable Banach spaces are homeomorphic one to another, right?) | |
| May 10, 2016 at 3:08 | comment | added | Włodzimierz Holsztyński | @MikhailOstrovskii -- your question would not specialize but complement my question. A topological characterization of injective metric spaces (or of a class) can be used to prove or disprove a homeomorphism theorem. | |
| May 10, 2016 at 2:35 | comment | added | Mikhail Ostrovskii | I would suggest to start with a concrete question: is there an injective metric space homeomorphic to an infinite-dimensional separable Hilbert space? For finite-dimensional Hilbert spaces and the Hilbert space of density character continuum the answer is clear: they are homeomorphic to $\ell_\infty^n$ and $\ell_\infty$, respectively. The second statement is a special case of the result of Toruńczyk, H. [Characterizing Hilbert space topology. Fund. Math. 111 (1981), no. 3, 247–262]. | |
| May 10, 2016 at 2:18 | comment | added | Paul Fabel | Is there a reasonable conjecture as to a plausible answer? ( For example the question at hand reminds me of the following: Are the topological spaces which underly length spaces precisely the metrizable, connected and locally path connected spaces?) | |
| May 10, 2016 at 1:46 | history | edited | Włodzimierz Holsztyński | CC BY-SA 3.0 |
cosmetic non-mth editing.
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| Oct 24, 2014 at 18:04 | comment | added | Włodzimierz Holsztyński | Monomorphism of the metric category of metric spaces are not isometric embeddings (they are just injective metric maps). Thus a pure categorist should take this into account. | |
| Oct 24, 2014 at 18:02 | comment | added | Włodzimierz Holsztyński | Isbell introduced (1) injective metric envelope, and (2) proved that the metric envelope of a Banach space is virtually the same as the Banach injective envelope (I rediscovered both a little later). | |
| Oct 24, 2014 at 17:54 | comment | added | Włodzimierz Holsztyński | @Ali, there are no non-trivial projective objects in the metric category of all metric spaces (only the empty space and singletons are projective). There are no non-trivial projective objects in the metric category of all bounded metric spaces. And in the metric category of all spaces of diameter $\le 1,\ $ a space $\ (X\ d)\ $ is projective $\ \Leftarrow:\Rightarrow\ d$ is the $\ 0\!-\!1$ metric. | |
| Oct 24, 2014 at 17:47 | history | edited | Włodzimierz Holsztyński | CC BY-SA 3.0 |
non-mathematical detail
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| Oct 24, 2014 at 16:44 | history | edited | Włodzimierz Holsztyński | CC BY-SA 3.0 |
typo
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| Oct 24, 2014 at 16:28 | history | edited | Włodzimierz Holsztyński | CC BY-SA 3.0 |
A clearer(?) question.
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| Oct 24, 2014 at 16:19 | comment | added | Ali Taghavi | @WłodzimierzHolsztyński Are there some known results about projective metric space with reversing the arrows in your definition? moreover, what is a relation with "functional analysis? I s it a good idea to translate this property for (commutative) separable $C^{*}$ ? | |
| Oct 24, 2014 at 16:14 | history | edited | Włodzimierz Holsztyński | CC BY-SA 3.0 |
A clearer(?) question.
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| Oct 24, 2014 at 15:54 | history | edited | Włodzimierz Holsztyński | CC BY-SA 3.0 |
typos
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| Oct 24, 2014 at 15:48 | answer | added | Tom Leinster | timeline score: 3 | |
| Oct 24, 2014 at 12:06 | comment | added | Andrej Bauer | Do you mean: "Characterize those topological spaces which are homeomorphic to an injective metric space?" or "Characterize those metric spaces which are injective, but only in terms of their topological properties?" | |
| Oct 24, 2014 at 12:02 | answer | added | Andrej Bauer | timeline score: 0 | |
| Oct 24, 2014 at 6:31 | history | edited | Włodzimierz Holsztyński |
metric spaces tag
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| Oct 24, 2014 at 6:16 | history | edited | Włodzimierz Holsztyński | CC BY-SA 3.0 |
necessary details missing (the same sense).
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| Oct 24, 2014 at 6:14 | comment | added | Włodzimierz Holsztyński | Just in case, don't worry about spaces which are not complete--they are never injective. A non-complete space is not a retract of its completion. | |
| Oct 24, 2014 at 6:10 | history | asked | Włodzimierz Holsztyński | CC BY-SA 3.0 |