Is a Space Connected if Its Closure Is Connected?

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Discussion Overview

The discussion revolves around the question of whether a space C is connected if its closure is connected. Participants explore this concept within the context of topology, specifically examining properties of connectedness and potential counterexamples.

Discussion Character

  • Debate/contested

Main Points Raised

  • One participant poses the initial question regarding the relationship between the connectedness of a space and its closure.
  • Another participant suggests that a counterexample may indicate that the answer is no, implying that the closure being connected does not guarantee the space itself is connected.
  • A different participant references the usual method for demonstrating that the circle is not homeomorphic to the real line, possibly hinting at related topological concepts.
  • Another participant prompts for examples of dense subsets of the real line, which may relate to the discussion of connectedness.
  • One participant reiterates the original question and hints at a straightforward counterexample involving the real line and a modification, though the details are incomplete.

Areas of Agreement / Disagreement

Participants do not reach a consensus, as there are competing views regarding the implications of connectedness in relation to closures, and the discussion remains unresolved.

Contextual Notes

Limitations include the incomplete nature of some contributions, such as the unspecified modification to the real line, which may affect the clarity of the counterexample being discussed.

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If the closure of a space C is connected, is C connected?
 
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What do you think?? Can you come up with a counterexample?? (this should suggest that the answer is no)
 
What is the usual way to show that the circle is not homeomorphic to the real line?
 
Can you think of some dense subsets of the real line?
 
blahblah8724 said:
If the closure of a space C is connected, is C connected?

There's a really easy counterexample. Just take the real line and ______ one single _______.
 

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