# Definition:Neighborhood of Infinity

## Topology

Let $X$ be a non-empty topological space.

A neighborhood of infinity is a subset of $X$ which contains the complement of a closed and compact subset of $X$.

## Real Analysis

### Neighborhood of Positive Infinity

A neighborhood of $+\infty$ is a subset of the set of real numbers $\R$ which contains an interval $\openint a \to$ for some $a \in \R$.

That is, a subset which contains all sufficiently large real numbers.

### Neighborhood of Negative Infinity

A neighborhood of $-\infty$ is a subset of the set of real numbers $\R$ wich contains an interval $\openint \gets a$ for some $a \in \R$.

That is, a subset which contains all sufficiently large negative real numbers.

## Complex Analysis

A neighborhood of $\infty$ in $\C$ is a subset of the set of complex numbers $\C$ wich contains a set of the form $\{z \in \C : |z| > r\}$ for some $r\in\R$.

That is, a subset which contains all complex numbers whose modulus is sufficiently large.