$X$ and $Y$ are homeomorphic. Show that also their one-point compactifications are homeomorphic.

Asked Mar 18 '16 at 01:47

Active Mar 18 '16 at 15:14

Viewed 1,533 times

Assume $X$ and $Y$ are two homeomorphic locally compact Hausdorff-spaces. Show that also their one-point compactifications are homeomorphic.

Give an example of two non homeomorphic locally compact Hausdorff-spaces but which one-point compactifications are homeomorphic.

Let $X$ and $Y$ be two locally compact Hausdorff-spaces and $f: X \rightarrow Y$ a homeomorphism. Define the two spaces one-point compactifications as $X'$ and $Y'$ and call the additional points $p$ and $q$.

Extend $f$ by letting $f(p) = q$.

To show that $f$ is a homeomorphism it is enough to show that $f(U)$ and $f^{-1}(V)$ are open in $Y'$ and $X'$ where $U$ and $V$ are open neighborhoods of $p$ and $q$, respectively.

This is the first problem i am solving which deals with one-point compactifications.

I am not sure how to show this and also have difficulties to come up with an example.

Any ideas?

edit:

First part is solved. Only the example they are looking for remains.

edited Jun 12 '20 at 10:38

Community

asked Mar 18 '16 at 01:47

JKnecht

6,543

1

Do you know the definition of the topology on the one point compactification? – Lee Mosher Mar 18 '16 at 02:01
1

@LeeMosher Yes i think i got it now. You can write the open set $U$ in $X'$ as ${p} \cup (X \setminus K)$ where $K$ is compact and then $f(U) = {q} \cup (Y \setminus f(K))$ is open. – JKnecht Mar 18 '16 at 02:37
Now only the example they are looking for remains. – JKnecht Mar 18 '16 at 02:38
4

How about $[0,1)$ vs. $[0,1) \sqcup (1,2]$? – quartermind Mar 18 '16 at 02:57
@quartermind yeah, that should do it i think. It was easier than i thought :) Thank you. – JKnecht Mar 18 '16 at 03:22

$X$ and $Y$ are homeomorphic. Show that also their one-point compactifications are homeomorphic.

0 Answers0

Linked