# Name for mappings that are “not quite projections”

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 $$Z$$.

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