How to show the one point compactification of a cylinder isn't a manifold.

Question

I'm trying to show that the one-point compactification of a cylinder $C^*$ = C$\cup${$\infty$} isn't a manifold.

The way I'm trying to show this is if $C^*$ is a manifold then if I take a neighborhood U of {$\infty$} there should be a subneighborhood V of U such that V\ {$\infty$} is connected. And then I'd have to show that V\ {$\infty$} is disconnected.

But I'm not sure if V\ {$\infty$} disconnected or not, my intuition says it should be I just can't prove it.

Thanks in advance

Answer 1

HINT: Let $S$ be a circle around $C$; $U=C^*\setminus S$ is an open nbhd of $\infty$. Show that no open nbhd of $\infty$ contained in $U$ can be connected, because it must meet both components of $C\setminus S$. Remember that open nbhds of $\infty$ have complements that are compact subsets of $C$.