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This may be a dumb question, but does anyone have an intuitive interpretation of when the set will be closed and when the set will be open, in the topological spaces?

I am confused especially when I heard that $\mathbb{Q}$ in $\mathbb{R}$ is neither an open nor a closed set. To me, I believe a set with a single element x, {x} should be a closed set, because we can also re-write it as [x, x].

I am very new to topology, so I am familiar with little theorems or assumptions in topology.

Thanks!

Caprikuarius
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  • A set with a single element ("singleton") is indeed closed (in the usual topology on $\mathbb{R}$; in other topologies things can get weirder). However, $\mathbb{Q}$ is not a singleton. So what's the connection? – Noah Schweber Sep 04 '19 at 19:48
  • I talks about the singleton because $\mathbb{Q}$ is the union of a countably infinite number of singletons. – Caprikuarius Sep 04 '19 at 19:49
  • Arbitrary unions of closed sets aren't necessarily closed. Every set is the union of all its singletons; does that mean that (in the context of $\mathbb{R}$ with the usual topology) every set is closed? Closedness is only guaranteed to be preserved by finite unions. Openness is preserved by arbitrary unions. – Noah Schweber Sep 04 '19 at 20:00

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