\begin{definition}[Definition:Continuous Real-Valued Function Space]
Let $X$ be a topological space.
Let $f : X \to \R$ be a continuous real valued mapping.
Then the set of all such mappings $f$ is known as '''continuous real-valued function space''' and is denoted by $\map \CC X$:
:$\map \CC X := \map \CC {X, \R} = \set {f : X \to \R}$
\end{definition}