Timeline for Extending a partially defined metric on a metrizable space
Current License: CC BY-SA 4.0
20 events
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| Nov 5, 2022 at 20:03 | vote | accept | omar | ||
| Nov 4, 2022 at 19:38 | history | edited | omar | CC BY-SA 4.0 |
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| Nov 4, 2022 at 19:15 | history | edited | omar | CC BY-SA 4.0 |
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| Nov 4, 2022 at 18:57 | comment | added | omar | @JoelDavidHamkins Thank you for your comments. I edited the question clarifying further my question. To clarify further. I don't want $A$ to be $Y\times Y$. But a case which is not far but interests me greatly is if one has a surjective continuous map $\pi:X\to Y$ and $A$ is the equivalence relation $x\sim y$ if and only if $\pi(x)=\pi(y)$. | |
| Nov 4, 2022 at 18:56 | history | edited | omar | CC BY-SA 4.0 |
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| Nov 4, 2022 at 18:05 | comment | added | Joel David Hamkins | Now I am confused about what the question means. You have set it up so that $A\subseteq X\times X$ is a set of pairs (not requiring that $A$ is the set of all pairs from some subspace $Y\subseteq X$). And you have stated that $d$ generates the topology on $A$, but of course $d$ is not a metric on $A$, so what is meant exactly? It would make sense if $d$ were defined on all pairs from a subspace. Is that what you mean? What exactly is the question? Are you asking: when does a metric realizing the subspace topology on a subspace extend to a metric realizing the whole (metrizable) space? | |
| Nov 4, 2022 at 16:12 | comment | added | omar | @JoelDavidHamkins, thank you for your comment. I modified the question accordingly. | |
| Nov 4, 2022 at 16:11 | history | edited | omar | CC BY-SA 4.0 |
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| Nov 4, 2022 at 15:04 | comment | added | Arno | With the locally compact second countable condition, we can modify Joel Hamkins counter-example by taking $X$ to be $\mathbb{N}$, $A = \mathbb{N}^2$ and $d$ a metric making $0$ the unique accumulation point. | |
| Nov 4, 2022 at 14:50 | comment | added | Joel David Hamkins | There is a trivial counter example to the question as stated. Let $X$ be the reals under the discrete topology, and take $A$ to be everything, defining $d$ as the Euclidean metric. This is continuous, closed, etc. in the discrete topology, but is already total and defines the wrong topology. You need to add the requirement that the partial metric $d$ defines the subspace topology on $A$, not just that the extension defines the topology on $X$. | |
| Nov 4, 2022 at 14:43 | answer | added | Joel David Hamkins | timeline score: 3 | |
| Nov 3, 2022 at 22:55 | history | edited | omar | CC BY-SA 4.0 |
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| Nov 3, 2022 at 22:52 | comment | added | omar | Yes, I require d to define the topology on X. Sorry the question wasn't clear I will edit it to make it clearer @YCor | |
| Nov 3, 2022 at 22:51 | comment | added | YCor | Do you require the extending metric $d'$ to satisfy any condition related to the topology? That $d'$ is continuous on $X\times X$? or, stronger, that $d'$ defines the topology of $X$? | |
| Nov 3, 2022 at 22:48 | comment | added | YCor | Related: mathoverflow.net/questions/431320/… | |
| Nov 3, 2022 at 22:48 | history | edited | YCor |
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| Nov 3, 2022 at 22:41 | history | edited | omar | CC BY-SA 4.0 |
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| Nov 3, 2022 at 22:40 | history | undeleted | omar | ||
| Nov 3, 2022 at 20:33 | history | deleted | omar | via Vote | |
| Nov 3, 2022 at 20:32 | history | asked | omar | CC BY-SA 4.0 |