Alexandroff Extension

Question

Sorry if this question sounds a little complicated/nooby, but I just want to make sure the way I am thinking about Alexandroff Compactification makes sense.

1. When we say that $X$ is a one point compactification of $Y$, what we are really saying is that by adding a single element to $X$, we can make (have made) $X$ compact. Correct?
2. Are one point compactifications unique?: Is there a one point compactification of a non-compact space $X$, up to homeomorphism?
3. The proposition which states that a homeomorphism from locally compact T$_2$ spaces, $X$, $Y$, can be extended to a homeomorphism between the Alexandroff Compactifications of $X$, $Y$ means what, intuitively? I think it means that if I can find a one point compactification of $X$, then it must be homeomorphic to a (the; depends on the answer to 2) one point compactification of $Y$.

I am using Armstrong, and the lack of explination leaves me wondering.

Answer 1

When we say that $X$ is a one-point compactification of $Y$, we’re saying that $Y$ is not compact, $X$ is compact and Hausdorff, and $|X\setminus Y|=1$. The one-point compactification is unique when it exists; see the answer to this question.

The statement that a homeomorphism $h:X\to Y$, where $X$ and $Y$ are locally compact and Hausdorff, can be extended to the one-point compactifications of $X$ and $Y$ simply means that if $p_X$ and $p_Y$ are the points added to $X$ and $Y$, respectively, in forming their one-point compactifications $X^*$ and $Y^*$, then the map

$$\tilde h:X^*\to Y^*:x\mapsto\begin{cases} h(x),&\text{if }x\in X\\ p_Y,&\text{if }x=p_X \end{cases}$$

is a homeomorphism.