Recall: given a (possibly non-$T_0$) topological space $X$, its Kolmogorov quotient $KX$ is the $T_0$ topological space formed by $X/\sim$ where $x\sim y$ if they are topologically indistinguishable. Denote the mapping $X\to KX$ of $x$ to its equivalence class $\pi$.
I have two loosely related terminology questions:
Is it okay to use the word "section" to refer to a mapping (and/or the image of such a mapping) $\gamma: KX \to X$ such that $\pi\circ \gamma = id$? (This would be the word from category theory, just wondering if there is another established terminology that is used.)
Is there a word for the equivalence relationship where two topological spaces $X$ and $Y$ are said to be equivalent if their quotients $KX$ and $KY$ are homeomorphic?
(I am particularly interested in the case where the non-$T_0$ topology comes from a semi-norm or a pseudo-metric; so if there is an answer to 2 when restricted to semi-normed spaces or pseudometric spaces, I'd be happy too.)
