TITLE: Can $X$ have compact connected components?
QUESTION [6 upvotes]: Let $X$ be a Hausdorff, locally compact but non-compact topological space.
If the (Alexandroff) one-point compactification is connected, can $X$ have compact connected components?

REPLY [3 votes]: I think I proved the following

Lemma
Let $X$ be a Hausdorff space and $C \subset X$ have a compact neighbourhood $K$. Then $C$ is a component of $X$ if and only if $C$ is a component of $K$.

in this answer.
For the present problem this implies a negative answer, since a compact set
in a locally compact Hausdorff space has a compact neighbourhood. Let $C$ be
a compact component of $X$, and let $K$ be some compact neighbourhood of $C$ in
$X$. Applying the lemma one way then shows that $C$ is a component of $K$, and
then applying it the other way shows that $C$ is a component of the
compactification, since $K$ remains a compact neighbourhood of $C$.