Prove that $K \subseteq \Bbb R$ is connected if for every $a,b \in K$ with $a

Question

If $K \subseteq \mathbb R$ is such that for every $a,b \in K$ with $a<b$, we have $[a,b] \subset K$, prove that $K$ is connected.

If $a,b \in K$ with $a<b$ such that $[a,b] \subset K$, then, for every $z \in \mathbb R$ with $a<z<b$, we know that $z \in [a,b] \subset K$, i.e. $z \in K$. This proves that $K$ is an interval.

We know that, in general, a subset $S \subseteq \mathbb R$ is connected if and only if $S$ is an interval. Therefore $K$ must be connected, since $K\subseteq R$ and $K$ is an interval.

Is this correct, or am I approaching this problem incorrectly?

score 1 · Accepted Answer · answered Nov 16 '16 at 14:51

1

Yes, if you know that an interval in $\Bbb R$ is connected, then your proof is fine. (As a side note, you've proven that a convex subset of $\Bbb R$ is connected.)

answered Nov 16 '16 at 14:51

Watson

23,793

That's an awesome side-note! :D. Not out of redundancy, but pure curiosity, how will I go about proving the theorem had I not known an interval in $\mathbb R$ is connected? I'm assuming a proof by contradiction might be easiest? :) – user860374 Nov 16 '16 at 14:58
@DJS : we can read about connected of intervals here, there, there, or there. – Watson Nov 16 '16 at 15:00
This can be generalized to linear continuum and order topology (see here). – Watson Nov 16 '16 at 15:02

Prove that $K \subseteq \Bbb R$ is connected if for every $a,b \in K$ with $a

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