This was a bonus question on a recent homework in my point-set topology class. I've since turned in the assignment but still don't have an answer to this question.
Let $X$ be a locally compact Hausdorff topological space. Let $X^+$ denote the one-point compactification of $X$. Prove or find a counterexample to the following claim: If $X^+$ is connected, then $X$ has no connected components which are compact.
The converse is true and straightforward to show. If we add the assumption that $X$ is locally connected, then the statement is true. And if we remove the requirement of local compactness, then it's false. But in its current form I haven't been able to make heads or tails of it.