# General Topology/Pointed spaces and support

Definition (pointed topological space):

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

Definition (morphism of pointed topological spaces):

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

Definition (support):

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

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

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