A metric space/topology question

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Discussion Overview

The discussion revolves around a problem related to compact metric spaces and their properties within the context of topology. Participants are addressing specific tasks involving compactness, closed sets, and connectedness in metric spaces, with a focus on proving certain properties and clarifying misunderstandings in the reasoning process.

Discussion Character

  • Technical explanation
  • Mathematical reasoning
  • Debate/contested

Main Points Raised

  • One participant presents a proof attempt for showing the existence of a point in a compact metric space that minimizes distance to a given point, but expresses uncertainty about the necessity of compactness in their reasoning.
  • Another participant suggests that the property of having convergent subsequences in compact metric spaces is relevant to the discussion.
  • A third participant points out that the limit of a sequence may not exist in closed sets, but does exist in compact sets, emphasizing the importance of compactness in the argument.
  • The original poster clarifies their reasoning after a typo is pointed out, attempting to formalize their argument regarding the distance between disjoint sets in a disconnected space.
  • Further elaboration is provided on the implications of disconnectedness in metric spaces, particularly concerning the existence of minimal distances between points in different components.
  • One participant expresses a desire to learn LaTeX or TeX for clearer communication of mathematical ideas.
  • The original poster indicates satisfaction with their understanding after further clarification and formalization of their arguments.

Areas of Agreement / Disagreement

Participants express varying levels of understanding and clarity regarding the problem, with some points of confusion remaining. There is no clear consensus on the sufficiency of the arguments presented, particularly regarding the necessity of compactness and the implications of disconnectedness.

Contextual Notes

The discussion includes assumptions about the properties of compact and closed sets, as well as the nature of distance in metric spaces. Some mathematical steps remain unresolved, particularly in the formalization of arguments related to disconnectedness.

Phillips101
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The question I'm doing is as follows:

(a) Show that every compact subset of a Hausdorff space is closed. I've done this.

(b) F is a compact metric space. F is a closed subset of X, and p is any point of X. Show there is a point q in F such that d(p,q)=infimum(d(p,q')) such that q' is in F.

Suppose that for every x and y in X there is a point m such that d(x,m)=0.5*d(x,y) and d(y,m)=0.5*d(x,y). Show that X is connected.

--

I don't really know how to do (b). This is my attempt at the first part of (b):

Take a sequence qn in F such that d(p,qn) decreases as n increases. Then by closedness of F, the limit of qn, say q.hat, is in F. This q.hat has the property that d(p,q.hat)=inf d(p,q') as required.

But, in the above solution, I didn't use the fact that X is compact? And I can't do the second part of (b) at all.

Thanks for any help!
 
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i don't quite understand the first part as it's written there but i think i can see that it would be useful to know that in a compact metric space any sequence has a convergent subsequence. for the second part the geometrical interpretation is that there is always a point halfway between x & y, so if X is *not* connected there are disjoint open sets (etc) & get a contradiction.
 
Because what you claim is not true of a closed set, the limit need not exist. However it is true for a compact set.

(Try to use a little Tex, it would make your question a lot more readable and more likely to be answered).
 
Thanks for the quick reply, and sorry, you're absolutely right the first part of (b) doesn't make sense as there's a typo. I'll make it clearer:

(b)X is a compact metric space. F is a closed subset of X, and p is any point of X. Show there is a point q in F such that d(p,q)=infimum(d(p,q')), where the infimum runs over all points q' in F.

Yeah, as you say, I thought that fact would be useful. But I don't really see any holes in my reasoning as it is *shrug*

And for the second part:
Suppose X is disconnected. Pick a connected component. This is then open and closed, as its complement is a union of open sets. Call this connected component F. Then, for every x in X\F, we have d(x,p)>0 for all p in F. So, there exists an x' and a p(x') such that this distance is minimal - ie d(x',p(x')) is the 'closest these two sets get'. And by (a), we see that p(x') is indeed in F, so d(x',p(x')) is strictly greater than zero.

I think that isn't sufficient, but I can't formalise it any more.
 
Last edited:
To some_dude, thanks for that clarification. And I haven't had the time to learn any Latex or Tex, but it is high on my to-do list. Thanks for the help
 
I've formalised everything and I'm very happy with the answers now - I appreciate all the help
 

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