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A subset $A$ of a topological space is called locally closed is it is open in its closure, or equivalently if it is the intersection of a closed set with an open set.

This terminology seems to be standard (e.g. Encyclopedia of Mathematics). However, I don't understand why this property has the name "locally closed".

  • Why "closed" and not "open"? Why not "locally open"?
  • Why the adverb "locally"? I know that, in general, a "local" property of a topological space $X$ is one where every $x\in X$ has a neighbourhood satisfying this property, but that doesn't seem relevant here (as every metric space is "locally closed" according to this definition, as we can just take the neighbourhood to be $X$).
Verónica Rmz.
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user1729
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    Not an answer, but a further link: locally closed set. For the Zariski topology a locally closed subset of $\Bbb P^n(K)$ is a quasiprojective variety. – Dietrich Burde Feb 14 '22 at 14:52
  • I think this will help you. – Moishe Kohan Feb 14 '22 at 14:52
  • @MoisheKohan Thanks. That certainly points out the similarity to the other "local" definition. Hmm. This suggests the definition: a subset $A$ is locally $P$ if every element in $A$ has a neighborhood $V$ in $X$ such that $A\cap V$ is has $P$ in $V$. But I haven't come across this definition of "local"-ness (yet...). – user1729 Feb 14 '22 at 15:28
  • Yes, this would be a good definition for "local versions" of other relative/extrinsic topological notions. I never saw this in other contexts though. (A potential example would be the definition of a "locally dense" subset.) – Moishe Kohan Feb 14 '22 at 16:04
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    ${ (x,0),x\in (0,1)}$ is locally closed in $\Bbb{R}^2$ – reuns Feb 15 '22 at 02:20
  • @reuns I'm not understanding your point. Could you elaborate a bit? – user1729 Feb 15 '22 at 10:23
  • I cannot guess what. The open unit interval $(0,1)$ embedded in $\Bbb{R}^2$ is locally closed. – reuns Feb 15 '22 at 10:50
  • @reuns Yes, but why is this relevant? – user1729 Feb 15 '22 at 11:37
  • If you want to define locality as it is more widely used, you need to something more specific: For all $x$ and every neighborhood of $x$ there is some smaller neighborhood of $x$ that has the property (and the "neighborhoods" need not be open - otherwise locally compact would be a far less useful condition). But the problem is, many well known uses of "local" do not quite satisfy that rule. For instance, locally connected would have to be changed to "locally open-&-connected", as this "locally connected" is connected-im-kleinen. – Paul Sinclair Feb 15 '22 at 18:56
  • @PaulSinclair Actually, in the case of the connected property it is a theorem that a space is "locally connected" iff it is "locally connected im kleinen" (see https://en.wikipedia.org/wiki/Locally_connected_space). (not the same thing at a particular point, but same thing when extended to every point in the space) – PatrickR Feb 12 '23 at 22:50

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