Interval exponential law and quotient homotopies

Evaluation $C_0(I,Y)\times I\to Y$ is continuous: at $(f,t)$ with $f(t)\in O$ open, choose a closed interval neighbourhood $J$ of $t$ contained in $f^{-1}O$. Then $W(J,O)\times {\mathrm{int}}_{I}J$ is a neighbourhood mapping into $O$. Relative interval neighbourhoods work at 0 and 1.

For continuous $H:X\times I\to Y$, the inverse image of $W(K,O)$ under $h(x)=H(x,-)$ is open: for any of its points $x$, the open set $H^{-1}O$ contains $\{x\}\times K$ and F1 supplies a tube. Thus h is continuous into $C_0(I,Y)$. Conversely compose $h\times {\mathrm{i}\mathrm{d}}_I$ with step 1.1. The resulting functions are inverse under evaluation at each $(x,t)$.

If a continuous $H:X\times I\to Y$ is constant on each fibre of $q\times {\mathrm{i}\mathrm{d}}_I$, its transpose is constant on q-fibres. By F2 it descends to a continuous $Q\to C_0(I,Y)$, which uncurries by step 2.1 to a continuous $Q\times I\to Y$. Apply this to the characteristic function of $U\subseteq Q\times I$ into the Sierpinski space with opens $\mathrm{\varnothing },\{1\},\{0,1\}$. Continuity of the characteristic function is exactly openness of U. If $(q\times \mathrm{i}\mathrm{d}{)}^{-1}U$ is open, the preceding descent makes U open. The map is surjective and continuous; hence it is quotient.

For CG spaces the kified correspondence is F4; the interval product already has its ordinary topology. Endpoint and basepoint equations are preserved pointwise by transpose and inverse transpose. Continuous maps from CG parameters landing in the corresponding subspace lift to its kification by the CG-source property contained in the conventions of F4. For WH targets these subspaces are closed and CGWH by F5; F3 supplies the closed-track quotient constructions. Applying the same correspondence to a cylinder parameter, or using the quotient-times-I conclusion of step 3.1, carries relative and based homotopies to relative and based homotopies.