Let A′ be 2ω1 with generalised Cantor space topology, and A′′ be 2ω1 with product topology, X a closed disc in a finite-dimensional Euclidean space. Then there is a continuous surjection A′→XA′′. I don’t know how to show that there is a topological space A with carrier set 2ω1 and a continuous surjection A→XA. Thanks to Alex Mennen for pointing out the problem.
However, because topology on A′ is finer than topology on A′′ here, this still shows how the proof of the Lawvere fixed point theorem can be applied here to give Brouwer fixed point theorem as corollary, which could conceivably be a publishable result (see what “Geometry and Topology” think about that), and this could still be sorta kinda maybe relevant to Scott’s original motivation for looking at the problem (if you’re okay with working with two different topologies on the space of agents, one finer than the other). But this is a very big space of agents you’re talking about here.
Correction: need not only that topology on A′ is finer than topology on A′′, but also, given arbitrary open subset of X, take pre-image under evaluation map in XA′′×A′′, projection onto first factor and then pre-image of that under the continuous surjection A′→XA′′, it needs to be shown that this set is open in both topologies. I believe that this can indeed be done for an appropriate class of spaces X for the pair of topologies in question.
Let A′ be 2ω1 with generalised Cantor space topology, and A′′ be 2ω1 with product topology, X a closed disc in a finite-dimensional Euclidean space. Then there is a continuous surjection A′→XA′′. I don’t know how to show that there is a topological space A with carrier set 2ω1 and a continuous surjection A→XA. Thanks to Alex Mennen for pointing out the problem.
However, because topology on A′ is finer than topology on A′′ here, this still shows how the proof of the Lawvere fixed point theorem can be applied here to give Brouwer fixed point theorem as corollary, which could conceivably be a publishable result (see what “Geometry and Topology” think about that), and this could still be sorta kinda maybe relevant to Scott’s original motivation for looking at the problem (if you’re okay with working with two different topologies on the space of agents, one finer than the other). But this is a very big space of agents you’re talking about here.
Correction: need not only that topology on A′ is finer than topology on A′′, but also, given arbitrary open subset of X, take pre-image under evaluation map in XA′′×A′′, projection onto first factor and then pre-image of that under the continuous surjection A′→XA′′, it needs to be shown that this set is open in both topologies. I believe that this can indeed be done for an appropriate class of spaces X for the pair of topologies in question.