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What's the difference between open and closed sets?

Especially with relation to topology - rigorous definitions are appreciated, but just as important is the intuition!

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    (seeded question)2010-07-28
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    This is relevant: http://mathoverflow.net/questions/19152/why-is-a-topology-made-up-of-open-sets2010-07-28
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    You should make your question more specific. Are you just looking for definitions here? Or do you want some intuition...2010-07-28
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    @BBischof: How's this?2010-07-28
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    Honestly, I think this question is a bit unfocused, but people are answering so it doesn't really matter.2010-07-28
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    I think this is too vague of a question and is not a very good seeded question. If the questioner knows what a topology is, then they would be able to make the question more precise. If they don't know what a topology is, then they shouldn't be asking this question.2010-07-28
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    The wording suggests that *open* and *closed* are mutually exclusive. Maybe you should ask ["How is a set different from a door?"](http://www.amazon.com/Topology-2nd-James-Munkres/dp/0131816292) (no, actually, don't ask that either.)2010-07-28
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    @Tom: that's intentional; I was hoping someone would point out a set can be both opened *and* closed. The [only answer that addressed this](http://math.stackexchange.com/questions/980/whats-the-difference-between-open-and-closed-sets/987#987), however, has problems with its definitions...2010-08-05

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