Topology Problem: Closure X Path-Connected?

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Discussion Overview

The discussion revolves around the question of whether the closure of a path-connected space is also path-connected. Participants explore this concept within the context of topology, examining examples and counterexamples related to connectedness properties.

Discussion Character

  • Debate/contested
  • Conceptual clarification

Main Points Raised

  • One participant questions if the closure of a path-connected space is also path-connected, suggesting it seems obvious but is uncertain about the proof.
  • Another participant counters the initial assumption, providing an example of the sine curve, which is path-connected but whose closure is not path-connected, indicating that the initial claim is false.
  • The same participant emphasizes the importance of remembering this counterexample, noting its frequent appearance in discussions of connectedness properties and exams.
  • A later reply mentions that the sine curve has a specific name, highlighting its significance in topology discussions.

Areas of Agreement / Disagreement

Participants do not reach a consensus; there is disagreement regarding the relationship between path-connectedness and the closure of a set, with one participant asserting it is false and providing a counterexample.

Contextual Notes

The discussion highlights the need for careful consideration of definitions and properties related to connectedness in topology, particularly regarding the closure of sets.

mveritas
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Hello, I have a question about topology.

If X is a path-connected space then is it also true that closure X is path-connected?

I think it's obvious, but I can't solve it clearly...
 
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Ah, this is a very good question. In fact, this very question was asked on my topology exam...

It is maybe obvious to you (and to me), but it is false. Consider

[tex]\{(x,\sin(1/x))~\vert~x\in ]0,1]\}\subseteq \mathbb{R}^2[/tex]

This is clearly a path connected set. However, it's closure is not path connected.

I suggest that you remember this example very well. Because it is a very frequent counterexample to all sort of connectedness-properties. And it's a popular exam question as well :smile:
 
micromass said:
Ah, this is a very good question. In fact, this very question was asked on my topology exam...

It is maybe obvious to you (and to me), but it is false. Consider

[tex]\{(x,\sin(1/x))~\vert~x\in ]0,1]\}\subseteq \mathbb{R}^2[/tex]

This is clearly a path connected set. However, it's closure is not path connected.

I suggest that you remember this example very well. Because it is a very frequent counterexample to all sort of connectedness-properties. And it's a popular exam question as well :smile:

-----------------------------------------------

Thank you!
 
It is used so often that it even has a (rather dull) name: the sine curve[/url].
 
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