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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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  • $\begingroup$ What is your motivation for considering open surjections in the first place? I'm just curious. $\endgroup$
    – David Carchedi
    Commented May 31, 2010 at 16:32
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    $\begingroup$ Open surjections, by a result of Moerdijk, are effective descent morphisms (apparently this is easier in the category of locales, or of topoi). Also, I was rereading a paper on topological stacks recently and they required a presentation by a topological groupoid where the chart map X_0->\X is an open surjection. A problem came up in my thesis whereby a map that should morally be a cover was only an open surjection, and only sometimes admitted local sections. To incorporate this apparent flaw I'd like to know better how the classes of maps interact. $\endgroup$
    – David Roberts ♦
    Commented Jun 1, 2010 at 1:07
  • $\begingroup$ I thought it would be something like this :-). Hmmm, ok. Is this a paper of Pronk? Anyhow, charts for topological stacks ALWAYS admit local sections. So, adding "is an open surjection" to the chart-condition only strengthens it. Similarly, the source and target maps of a topological groupoid both have the unit map as a GLOBAL section. So, saying that they should be open only ADDS to the condition. Where you'd run into trouble would be with principal-bundles I presume? $\endgroup$
    – David Carchedi
    Commented Jun 1, 2010 at 1:47
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    $\begingroup$ A paper of Pronk? No. And the condition of the chart for the stack was only that it had to be open surjection (I've got to track it down now, I know :-) I'm not sure where the principal bundles come into it - I mean I know about stackifying a topological groupoid by considering the stack of principal bundles for it, but that's not I'm looking at. The stack reference is misleading, perhaps. I'm really only interested in the topology, with an eye to saying the open surjection pretopology is equivalent to the open cover pretopology for such-and-such spaces and maps. $\endgroup$
    – David Roberts ♦
    Commented Jun 1, 2010 at 6:28
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    $\begingroup$ Such a full subcategory can't have $\mathbb R^2$ in it: the complex squaring map is open and has no section in a neighborhood of $0$. $\endgroup$
    – Tom Goodwillie
    Commented Mar 15, 2012 at 3:09

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Maybe this is not quite satisfactory answer, but the category of zero-dimensional Polish spaces and their continuous maps has the required property: each open continuous map between Polish zero-dimensional spaces has a section (this follows from a suitable Michael's Selection Theorem, see e.g. http://www.renyi.hu/~descript/papers/Michael_monthly.pdf).

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