Describing a one-point compactification

Question

Show that $X = S^{1} \cup ((0,2) \times \{0\}) \subset \mathbb{R}^{2}$ is locally compact and find its one-point compactification.

Definitions

$S^{1}$ is the open unit circle. Our definitions are these -

A space $X$ is compact if a finite subcover can be extracted from every open cover $\mathcal{U}$ of $X$. A space $X$ is locally compact if every element $x \in X$ admits a compact neighbourhood.

A one-point compactification of a topological space $(X, T)$ is a compact Hausdorff space $(\tilde{X}, \tilde{T})$ together with an embedding $i: X \rightarrow \tilde{X}$ with the property that $|\tilde{X} - X| = 1$.

My questions

Im not too sure about my proof of the local compactness of X:

Let $x \in X$, and let $\mathcal{U}$ be an open cover of a subspace $A \subset X$ containing $x$. Then there exists an element $U \in \mathcal{U}$ so that $x \in U$. Choose $\delta_{U}$ so that $B(x, \delta_{U}) \subset U$. Then $B(x, \delta_{U})$ is an open neighbourhood of $x$ and $U$ is its finite subcover.

In fact, reading it again, it feels wonky. Im still trying to develop an intuition for the notion of compactness. If it is indeed not correct, how would I start proving $X$ is locally compact?

Furthermore, I have no idea how to describe the one-point compactification of $X$. Ive made a drawing of $X$ and I think the final result should be homeomorphic to a figure resembling an 8, but how would I intuitively find this? And how would I then describe it rigorously?

Thanks in advance!

Answer 1

HINT: To show local compactness, show that each point of $X$ has a nbhd that is a closed, bounded subset of $\Bbb R^2$ and therefore is compact. (In fact every point of $X$ except $\langle 1,0\rangle$ has a nbhd homeomorphic to $[0,1]$.)

You are correct in thinking that $\tilde X$ is homeomorphic to a figure $8$; in effect the point at infinity turns the open segment $(0,2)\times\{0\}$ into a copy of $S^1$. Here’s a suggestion for a more rigorous writeup. First let

$$C=\{\langle x,y\rangle\in\Bbb R^2:(x-2)^2+y^2=1\}\;,$$

the circle of radius $1$ centred at $\langle 2,0\rangle$, let $C_0=C\setminus\{\langle 3,0\rangle\}$, and let $Y=S^1\cup C_0$.

• Show that $X$ is homeomorphic to $Y$; for instance, map $(0,1)\times\{0\}$ nicely to the open upper half of $C_0$ and $(1,2)\times\{0\}$ to the open lower half.
• Then show that $\tilde X$ is homeomorphic to $S^1\cup C$.