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Prove or Disprove

Any convex and closed set in R has to be a closed interval of the form [a,b]

I initially thought that R itself is convex and closed and not in this form. So, counter-example.

But then I realized that R is also open, that is to say R is a clopen set. Then this approach may fail, right?

John Stern
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  • Indeed, $\mathbb R$ is a counterexample. It does not matter that $\mathbb R$ is also open. – gerw Sep 04 '17 at 09:09

1 Answers1

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Any bounded convex closed set has to be a closed interval of the form $[a,b]$ (proof: let $a$ be the infimum of the set and $b$ the supremum; bounded implies both are in $\mathbb R$; closure implies both are in the set; convex implies every point inside the interval is too).

As you say, unbounded sets such as $\mathbb R$ itself are not of this form - if you want an example which is not open then $[a,\infty)$ or $(-\infty, b]$ will do.

Especially Lime
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