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Is there a known name for the following definition?

Consider topological spaces $X$, $Y$ and $f: X \rightarrow Y$ a continuous mapping. Then, $f$ is an "almost projection" if there is a topological space $Z$ and a continuous onto mapping $g: Y\times Z\rightarrow X$ s.t. $f\circ g$ is equal to the canonical projection from $Y\times Z$ to $Y$.

For example, given any topological space $W$, the canonical mapping from the cone $X=CW$ to $Y=[0,1]$ is an "almost projection": let $Z=W$ and $g:[0,1]\times W\rightarrow CW$ be the canonical mapping from the cylinder to the cone.

On the other hand, the mapping from $X=\{(x,y)\in\mathbb{R}^2\mid xy=0\}$ to $Y=\mathbb{R}$ defined by $f(x,y)=y$ is not an "almost projection": for any $y\in\mathbb{R}\setminus0$ and $z\in Z$, we must have $g(y,z)=(0,y)$ since $f^{-1}(y)=\{(0,y)\}$, by continuity of $g$ we get that that for any $y\in\mathbb{R}$, $g(y,z)=(0,y)$, but then $g$ is not onto since $(1,0)\in X$ is not in the image.

A possible variant of this concept can be obtained by replacing $Y\times Z$ by an arbitrary fiber bundle over $Y$ with fiber $Z$.

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  • $\begingroup$ Something involving the word 'dominated'? For instance, in the category of spaces over $Y$, $f$ is dominated by a projection? $\endgroup$ – David Roberts Dec 13 '18 at 23:20

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