# Definition A [[Topological Space]] $(X, \mathcal{T})$ is a *Hausdorff space* or *$T_2$ space* if, for $x, x' \in X$, there always exist [[neighborhood|neighborhoods]] $U_x$ of $x$ and $U_{x'}$ of $x'$ such that $U_x \cap U_{x'} = \emptyset$. All [[Metric Space|metric spaces]] are Hausdorff. In physics, we always assume topological spaces are Hausdorff spaces. # $\mathbb{R}^n$ The idea that a line joining any two points of $\mathbb{R}^n$ ([[Real Coordinate Space|R^n]]) can be infinitely subdivided can be made more precise by saying that any two points of $\mathbb{R}^n$ have non-intersecting [[Neighborhood|neighborhoods]]. They will also have [[neighborhood|neighborhoods]] that do intersect, but if the neighborhood radius $r$ is small enough, we can make them disjoint.