I am trying prove the following theorem:(adsbygoogle = window.adsbygoogle || []).push({});

If a set [tex]A[/tex] is connected, and if [tex]A \subseteq B[/tex] and [tex]B \subseteq cl(A)[/tex] (where cl(A) is the closure of A), then [tex]B[/tex] is connected.

Just so everyone is on the same page, a set X is connected if the only subsets of X that are both open and closed are the empty set and X itself. I gave the definition in case that isn't the most common definition. I'm sure there are a number of ways of saying it.

I thought I had something going, and all I needed was to show that if a set is open, then the interior of its closure is the set itself (hence my last post). Of course this is false: take any bounded open interval on the real line and remove one point from the middle. Any suggestions?

thanks

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# A point-set theorem

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