Definition:Continuous Real-Valued Function Space

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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 $\map C X$:

$\map C X := \map C {X, \R} = \set {f : X \to \R}$