Mapping path space replacement of a map

For a continuous map $f:X\to Y$, let $Y^I=C_0(I,Y)$ with the ordinary compact-open topology. Define $E_f=\{(x,\gamma )\in X\times Y^I:\gamma (0)=f(x)\},p_f(x,\gamma )=\gamma (1),j_f(x)=(x,c_{f(x)}),$ where $c_y$ is the constant path with value $y$. These use the product and subspace topologies of A map into a product is continuous iff each of its components is; the projections are continuous and open; and each projection is surjective when every factor is nonempty, which for an infinite index set uses the Axiom of Choice and Subspace topology: the traces of the open sets, its closed sets and its bases, the continuity of the inclusion, and the characteristic property of a map into a subspace. They are called the mapping-path space, its endpoint projection, and its constant-path inclusion.

Evaluation is continuous by Interval exponential law and quotient homotopies, so $p_f$ is continuous. The constant-path map $x\mapsto c_{f(x)}$ is continuous by transposing $(x,t)\mapsto f(x)$; pairing with $x$ and restricting to $E_f$ shows that $j_f$ is continuous. Also let $r_f(x,\gamma )=x$; this is continuous as a restricted product projection. No arbitrary selection of paths is part of these definitions.

For CGWH spaces use the kified path space, k-product and kified indicated subspace instead; the same interval exponential law supplies these maps. Empty $X$ gives empty $E_f$. If $Y$ is empty then the existence of $f$ already forces $X$ empty. For a one-point domain this is the space of paths starting at its specified image; for a one-point target it identifies with $X$. The factorization and homotopy claims are proved next.