> [!definition] > > Let $(X, \topo)$ be a [[Topological Space|topological space]] and $f \in C(X)$ be a real/complex-valued [[Space of Continuous Functions|continuous function]]. The **support** of $f$, > $ > \supp{f} = \ol{\bracs{x: f(x) \ne 0}} > $ > is the [[Topological Closure|closure]] of points where $f$ is non-zero. In other words, it is the smallest [[Closed Set|closed set]] outside which $f$ vanishes. > > The support being closed allows separation of points outside from it using [[Neighbourhood|neighbourhoods]].