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For pointed topological spaces $(X,x_0)$ and $(Y,y_0)$, the pointed mapping space ${\mathrm{Map}}_{\ast }(X,Y)$ 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, ${\mathrm{Map}}_{\ast }(S^1,Y)$ 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 ${\mathrm{Map}}_{\ast }(S^n,Y)$ 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.

A standard form of the Sullivan conjecture, proved by Haynes Miller, states that for a finite group $G$ and a connected finite-dimensional CW complex $X$, the pointed mapping space from the classifying space of a discrete group $BG$ to $X$ is a weakly contractible space:Thus every basepoint-preserving continuous map $BG\to X$ is null-homotopic, and higher homotopy families of such maps are trivial as well. Miller's Theorem A allows the more general assumption that every finitely generated subgroup of $G$ is finite.^[ref]

The finite-dimensional target hypothesis is substantial. For $G=\mathbb{Z}/2\mathbb{Z}$, maps from $BG$ to a finite-dimensional sphere satisfy the theorem, while the identity map of $BG$ is not null-homotopic: it induces the identity on its nontrivial fundamental group. That target $BG$ is infinite-dimensional. The theorem concerns these unstable mapping spaces; it cannot simply be transferred to every stable mapping construction.