- #1

ProfuselyQuarky

Gold Member

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## Main Question or Discussion Point

I feel like I ask too many questions here, so I'm sorry about that. But, anyway, what is a clopen set? I was watching something the other day that a friend sent me, and, out of the blue, the guy starts referring to a set as being "clopen" with no explanation. I tried to break all the definitions I know down into little bits, but I still find it confusing. Apparently a clopen set is both open and closed.

From Wiki:

When it says "which leaves the possibility of an open set....." are they talking about a completely different set or something related to the initial closed set...?

Obviously I am wrong and lost, so clarification would be greatly appreciated. I am going in circles.

*How??*Letting ##A\subseteq \mathbb{R}##, I know that a set A is open if every point of A is an interior point of A . A is closed, however, if and only if ##\mathbb{R}\setminus A##.From Wiki:

I know that a complement of a set is all the things outside of the set, but I just don't understand. It's not the (singular) set itself that's both open and closed, right? All I get out from reading is that you can have a closed set with an open complement and an open set with an open complement. I can't see how that makes a set clopen. It is still either open or closed.A set is closed if its complement is open, which leaves the possibility of an open set whose complement is also open, making both sets both openandclosed, and therefore clopen.

When it says "which leaves the possibility of an open set....." are they talking about a completely different set or something related to the initial closed set...?

Obviously I am wrong and lost, so clarification would be greatly appreciated. I am going in circles.