Closed sets in a topological space

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Discussion Overview

The discussion revolves around the properties of closed sets within subsets of a topological space, specifically whether closed subsets of a subset A are also closed in a larger subset B when A is contained in B. The scope includes theoretical considerations of topology and the implications of compactness and closure.

Discussion Character

  • Debate/contested
  • Technical explanation
  • Conceptual clarification

Main Points Raised

  • Some participants question whether any closed subset of A is also a closed subset of B, given that A is a subset of B.
  • It is noted that A is a closed subset of itself, but this does not necessarily imply that closed subsets of A are closed in B.
  • One participant suggests that if A is compact in B, then the statement about closed subsets might hold true.
  • Another participant counters that simply insisting A is closed is sufficient, and that compactness does not imply closure in all topological spaces.
  • There is confusion regarding the relationship between compactness and closure, with one participant expressing surprise at the assertion that compact does not imply closed.
  • A specific example of the Zariski topology is provided to illustrate that a compact set can be open but not closed.

Areas of Agreement / Disagreement

Participants express differing views on the necessity of compactness versus closure for the original statement to hold. There is no consensus on the implications of compactness in this context, and the discussion remains unresolved regarding the relationship between closed subsets and their properties in larger sets.

Contextual Notes

Participants acknowledge that the definitions of closed and compact sets may vary depending on the topology in question, and there are unresolved assumptions about the nature of the topological spaces being discussed.

logarithmic
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If A\subseteq B are both subsets of a topological space (X,\tau), is it true that any closed subset of A is also a closed subset of B?
 
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logarithmic said:
If A\subseteq B are both subsets of a topological space (X,\tau), is it true that any closed subset of A is also a closed subset of B?

Note that A is a closed subset of A...
 
Hurkyl said:
Note that A is a closed subset of A...
Hmm, I'm not sure. I never said A was closed. A is in B which is in X. And some other set in A, maybe call it U, is closed. I think the answer would be U is closed in X but I don't quite see why.
 
Hurkyl's point is that any topological space is both open and closed as a subset of itself. If A is NOT closed as a subset of topological space B, since it IS closed as a subset of itself, the statement "any closed subset of A is also a closed subset of B" is false.
 
HallsofIvy said:
Hurkyl's point is that any topological space is both open and closed as a subset of itself. If A is NOT closed as a subset of topological space B, since it IS closed as a subset of itself, the statement "any closed subset of A is also a closed subset of B" is false.
Ahh i see. Thanks. But is that the only problem here? If we insist that A is compact in B, then that fixes the problem and the statement is true, right?
 
Um, just insisting that A is closed is all you need, nothing to do with compactness. In fact compactness won't help you at all - compact does not imply closed (e.g. the Zariski topology on R).
 
?? I thought compact did imply closed!

Oh, I see. I started to give the proof and then realized I was saying "given points p and q construct neighborhoods about p and q that do not intersect". That's not possible in some topological spaces.
 
If you don't know what the Zariski topology is (and Halls does but forget, temporarily) consider the topology on R given by:

U is open if and only if U contains the interval (0,1) - the set (0,1) is in this and is certainly compact, but not closed.
 

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