The class of open surjections $Q \to X$ is a Grothendieck pretopology on the category $Top$ of spaces, and includes the class of maps $\amalg U_\alpha \to X$ where $\{U_\alpha\}$ is an open cover of $X$. There are large classes of spaces for which these two pretopologies aren't equivalent (any non-locally contractible space, for example). What I'd like to know is if there are any spaces for which they are. My question can be split into two parts:

Is there a full subcategory of $Top$ such that every open surjection admits local sections?

Is there a non-full subcategory, like that of finite CW complexes and cellular maps (not that I'm claiming this is), in which every open surjection - in this category - admits local sections?

Clearly the non-full subcategory part needs to include enough maps to be sensible, e.g. every continuous map is homotopic to one in the subcategory, and enough objects to also be considered nontrivial.

Edit: I'm bumping this question because it received little interest, and I thought I'd explain the example which brought me to this idea.

Consider the *path fibration* $P_xX \to X$, the total space of which is the space of based paths with the compact-open topology. If $X$ is locally contractible then this is fibre homotopy trivial, and in particular admits local sections. As $P_xX$ is contractible, it can be seen as a sort of 'free resolution' of the space $X$ - some sort of 'cover'. (If we work in the smooth setting, and let $X=G$ a Lie group, then $P_eG \to G$ is even a locally trivial $\Omega G$-bundle.) However, going to the other extreme and only asking for $X$ to be path connected and locally path connected then $P_xX \to X$ is an open surjection. There are lots of other maps which are open surjections which admit local sections (such as principal bundles)

I'm happy for open surjections to be covers (i.e. form a Grothendieck pretopology) but then I want to be able to specify when open surjections and open covers generate the same Grothendieck topology, and this involves finding a category of spaces (which I'd like to be enough to model all homotopy types and mapping spaces correctly) where open surjections admit local sections.

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