Does this condition on a continuous surjection have a name?

Question

Let $\pi : Y \to X$ be a continuous surjection of topological spaces. Does the following condition on $\pi$ have a name?

For every $x \in X$, every open set $U$ containing $x$, and every $y \in \pi^{-1}(x)$, there exists a local section $s : U \to Y$ of $\pi$ (a map such that $\pi \circ s$ restricts to the identity on $U$) such that $s(x) = (s \circ \pi)(y) = y$.

For example, every (locally trivial) fiber bundle has this property, but there are more general examples, such as the projection of the circle $S^1 \subset \mathbb{R}^2$ onto, say, the $x$-axis.

Answer 1

In paragraph 2.7 of

George Janelidze and Walter Tholen, How algebraic is the change-of-base functor? Lecture Notes in Mathematics Volume 1488, 1991, pp 174-186,

these maps are called local split-epimorphisms or locally sectionable.

As examples they mention local homeomorphisms, covering maps, or the projection $X \to X/G$ where $X$ is completely regular $G$-space with $G$ a compact Lie group. They also point out that a local split-epimorphism need not be open, e.g. the projection $(-1,1]\to \mathbb{R/Z}$ or surjective polynomial functions $\mathbb{R \to R}$.

One of the main results of the paper (Theorem 2.7) asserts that for a local split epimorphism $p \colon X \to Y$ the change-of-base functor $p^\ast\colon \mathsf{Top}/Y \to \mathsf{Top}/X$ is monadic.

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