# Definition When we use a [[Neighborhood#^8771d7|distance function]] $d(\pmb{x}, \pmb{y})$ to define [[Neighborhood|neighborhoods]] and thus [[Open Set|open sets]] in $\mathbb{R}^n$, then we say that $d(\pmb{x}, \pmb{y})$ induces a topology on $\mathbb{R}^n$ In this sense, the distance function allows us to $d(\pmb{x}, \pmb{y})$ make $\mathbb{R}^n$ into a [[Topological Space|topological space]]. Note that the induced topology does **not** depend on the specific form of $d(\pmb{x}, \pmb{y})$. All we need to know is only a notion that the distance between points can be made arbitrarily small and that no two distinct points have zero distance between them