Topology Problem: Closure X Path-Connected?

[mveritas]

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...

 

[micromass]

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

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

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

 

[mveritas]

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

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

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

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Thank you!

 

[Landau]

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