Definition:Continuous Real-Valued Function Space

Definition
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 $\CC \paren {X}$:


 * $\CC \paren {X} := \CC \paren {X, \R} = \set {f : X \to \R}$