Is there a name for spaces that always have local sections?

Question

Given a continuous map $p:E \rightarrow B$

Suppose for every point $b \in B$ and a point $x \in p^{-1}b$ in the fibre of it, there is an open set $V$ of $B$ that contains the point $b$ such that there is a local section of $p$ from $V$ and that goes through $x$.

Is there a name for a standard name for spaces with this property?

Etale spaces, it seems to me, always have this property; but I don't think they are characterised by it.

Answer 1

In the category of smooth manifolds you get a nice answer: You class of maps equals the class of submersions. But this fails in TOP and PL category if E has dimension 2 or more.

Here is another example to consider: the map $p(x,y)=x+y$, restricted to the algebraic subset $xy=0$: it satisfies your condition.