I have some trouble with following problem:
Let X be a locally compact Hausdorff space where $X^+=X \cup \{\infty\}$ is its one-point compcatification and $Y \subseteq X$. Compute the closure of Y in $X^+$ in terms of its closure in X.
This is what I have so far:
The adherent points of Y which are elements of X are still adherent points of Y in $X^+$, so the question is whether the point at the infinity is an adherent point of Y in $X^+$ or not. Consider two cases:
- Case 1: The set Y is a subset of some compact K included in X.
- Case 2: Case 1 does not hold, that is, for any compact K included in X there are points of Y which are not in K.
Heres what I need help with:
- I Think I just need to decide what happens with the point at the infinity in each case, but I am a little confused about that part?
- Is the start correct?
$\infty$ belongs to the closure of $Y$ in $X^{+}$ iff $K^{c}$ intersects $Y$ for every compact set $K$ iff $Y$ is not a subset of any compact set $K$ in $X$. This is so iff $Y$ is not relatively compact set in $X$.