Neighborhoods of $\infty$ in the extended plane $\mathbb{C}\cup\{\infty\}$

Question

This is just a definition problem. Could somebody help me with the usual way to define the neighborhoods of $\infty$ on the extended plane $\mathbb{C}\cup\{\infty\}$?

Any kind of info/bibliography would be appreciated!

Answer 1

The open sets around $\infty$ are generated by the open "balls"

$$ B_R = \{x : |x| > R\}. $$

Thus a set $A \subseteq \mathbb{C} \cup \{\infty\}$ is a neighborhood of infinity if and only if there exists some $R$ such that $|x| > R$ implies $x \in A$.

Answer 2

This is an example of the one-point compactification. The open sets containing $\infty$ are those of the form $\{\infty\}\cup K^c$ (where "$X^c$" denotes the complement of $X$), for $K$ a compact (in the usual sense) subset of $\mathbb{C}$.

Something worth keeping in mind: if $U\subseteq\mathbb{C}\cup\{\infty\}$ is open, then so is $U\setminus\{\infty\}$. However, if $V\subseteq \mathbb{C}$ is open in the usual sense, it is not necessarily true that $V\cup\{\infty\}$ is open. For example, let $H$ be the open upper half plane; then $H\cup\{\infty\}$ is not open. If $V\subseteq \mathbb{C}$ is open in the usual sense, then $V\cup\{\infty\}$ is open iff $V$ is all of $\mathbb{C}$ outside some ball of finite radius.