= Pointed mapping space
{title2=$\operatorname{Map}_*(X,Y)$}
For <pointed topological spaces> $(X,x_0)$ and $(Y,y_0)$, the <pointed mapping space> $\operatorname{Map}_*(X,Y)$ consists of <continuous maps> $f:X\to Y$ with $f(x_0)=y_0$. It carries the subspace <topology> from the <compact-open topology> on the ordinary mapping space, commonly followed by <kification> when working with <compactly generated spaces>. Its basepoint is the constant map with value $y_0$.
For the pointed <circle>, $\operatorname{Map}_*(S^1,Y)$ is the <loop space> $\Omega Y$: a circle map is a path returning to its starting point. More generally, for well-behaved pointed spaces, path components of $\operatorname{Map}_*(S^n,Y)$ correspond to $\pi_n(Y,y_0)$. Thus a mapping space turns families of maps and <homotopies> into points and paths of another <topological space>. The choice of mapping-space <topology> ensures that this correspondence works continuously.
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