Does Every Open Set Contain a Compact Set?

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Discussion Overview

The discussion revolves around whether every open set contains a compact set, exploring the implications and examples related to this question within the context of set theory and topology.

Discussion Character

  • Debate/contested

Main Points Raised

  • One participant asserts that every open set contains a compact set, referencing the empty set as an example.
  • Another participant challenges this by stating that demonstrating one open set contains a compact set does not address the original question posed by the OP.
  • A different viewpoint suggests that if an open set is non-empty, it contains a singleton set, which is compact due to being finite.
  • One participant emphasizes that every open set contains the empty open set, which in turn contains the empty compact set, suggesting a broader interpretation of the question.

Areas of Agreement / Disagreement

Participants express differing views on the interpretation of the original question, with some agreeing on specific cases (like the empty set) while others argue that these examples do not sufficiently address the general case.

Contextual Notes

The discussion highlights the need for clarity regarding definitions of open and compact sets, as well as the implications of including the empty set in the argument.

AxiomOfChoice
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Is it true that every open set contains a compact set?
 
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Well, the empty set is open. And yes, even the empty set contains a compact set.
 
Showing that one open set contains a compact set does not prove what the OP wanted!

However, I can finish by showing the "easy" part: If A is not empty, then it contains, say, "a". The singleton set "a", since it is finite, is compact.
 
My point was, that every open set contains the empty open set, which contains the empty compact set. So I was not really doing *one* open set after all.
 

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