# Definition ## General Topological Spaces Suppose $\mathcal{T}$ gives a topology to a [[set]] $X$, i.e., $(X, \mathcal{T})$ is a [[Topological Space]]. $N$ is said to be a *neighborhood* of a point $x \in X$ if $N \subset X$ and $N$ contains at least one $U_i \in \mathcal{T}$ such that $x \in U_i$. If $N$ is an [[open set]], then we say it is an open neighborhood. ## Metric Spaces Let's take $\mathbb{R}^n$ ([[Real Coordinate Space|R^n]]) as an example. Defining a [[Metric]] between two points $\pmb{x} = (x_1, x_2, \ldots, x_n)$ and $\pmb{y} = (y_1, y_2, \ldots, y_n)$ such that $\pmb{x}, \pmb{y} \in \mathbb{R}^n$ ([[Real Coordinate Space|R^n]]) $ d(\pmb{x}, \pmb{y}) = [(x_1 - y_1)^2 + (x_2 - y_2)^2 + \ldots + (x_n - y_n)^2]$ The neighborhood of radius $r$ of the points $\pmb{x}$ in $\mathbb{R}^n$ is the set of points $N_r(\pmb{x})$ whose distance from $\pmb{x}$ is less than $r$. See the figure below for an example in $\mathbb{R}^2$, where the interior of the circle is the neighborhood defined by the distance function above. The circle itself is not included. ^8771d7 ![[Pasted image 20210122192044.png]]