created topological submersion. I’ve seen more than one definition of this, and both could be useful. My natural inclination is to the more general, where each point in the domain has a local section through it.

On a side note I use a related condition in my thesis for a topological groupoid over a space: every object is isomorphic to one in the image of a local section. This was used in conjunction with local triviality of topological bigroupoids to define certain sorts of 2-bundles.

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