Subset is Open and Closed implies Subset is Entire Space

Question

I'm reading a proof about the maximum principle. In the proof they assume $D \in \mathbb{R}^N$ is an open, bounded, connected domain. Then they construct an $M \subset D$ which they show is non-empty, open, and closed. They the immediately conclude that $M = D$. I suspect this is some topological fact, but I'm not sure how to arrive at this conclusion. Any help?

Answer 1

You can see that this is true since $D$ is connected.

If $M \subsetneq D$ then it follows that $M^c:=D \setminus M$ is open as well, since $M$ was closed. Now $M^c \cup M$ form a partition of $D$, contradicting connectedness.

Answer 2

In topology a space is connected if and only if the only subsets which are clopen are the whole domain and the empty set.

https://en.wikipedia.org/wiki/Clopen_set

The Euclidean spaces are easily shown to have this property; so a general strategy when proving an entire space has some property is to show that the set of things which have the desired property is both open and closed (and non-empty... show at least one thing exhibits the property in question).