Non-empty intersection between a compact and an unbounded connected subset of $\mathbb{R}^d$

Question

I am quoting from MathOverflow, where I have just read it as part of a comment: "If $C$ and $S$ are, resp., a compact and a connected unbounded subset of $\mathbb{R}^d$ such that $C\cap S \ne \emptyset$, then $S$ meets $\textrm{bd}(C)$." As far as I understand, $\textrm{bd}$ means "boundary".

I have my own proof of this, but the statement looks like one of those basic results in topology that you are going to use in a number of situations. So I wonder if anybody could give a reference.

In addition, what about some generalizations?

Answer 1

This is a special case of "you cannot connect the inside to the outside without crossing the boundary". As it is stated in Elementary Topology. Textbook in Problems

12.26 Let $F$ be a connected subset of a space $X$. Prove that if $A \subset X$ and neither $F \cap A$, not $F \cap (X\setminus A)$ is empty, then $F \cap \operatorname{Fr} A \ne \emptyset$.

Sketch of (my) proof: If $F \cap \operatorname{Fr} A = \emptyset$, then $F \cap \operatorname{Int} F$ and $F \cap \operatorname{Ext} F$ form a separation of $F$.

So in the case you describe the only relevant consequence of the facts that $C$ is compact and $S$ is unbounded, is that $S \not\subset C$. The fact that the space is $\mathbb{R}^n$ is not needed at all.

Answer 2

As pointed out by Niels Diepeveen in his answer, this is a special case of something more general, which amounts to Proposition 3 in Chapter I, Section II.1 of N. Bourbaki's Elements of Mathematics - General Topology, Part 1 (p. 109 in the 1966 edition by Hermann).