General Topology/Pointed spaces and support

Definition (pointed topological space):

A pointed topological space is a pair $(X,x_{0})$ where $X$ is a topological space and $x_{0}\in X$.

Definition (morphism of pointed topological spaces):

A morphism of pointed topological spaces $(X,x_{0})$, and $(Y,y_{0})$ is a continuous function $f:X\to Y$ such that $f(x_{0})=y_{0}$.

Definition (support):

Let $f:Z\to X$ be a continuous map, where $x_{0}\in X$ is a distinguished point. Then the support of $f$ is defined to be

$\operatorname {supp} f:={\overline {X\setminus f^{-1}(x_{0})}}$.

Often, $X$ is a topological magma with identity, and $x_{0}$ is the identity. For instance, $X=\mathbb {R}$ and $0\in \mathbb {R}$ would be a possibility.