Arguing against Choice, understood internally, applies nonmonotonic logic to restrict what can be deduced. It is fundamentalist in its rejection of sophisticated reasoning supporting clear thinking. (Sorry if that's too cryptic, happy to explain privately if this worries you.)
There's a sense in which by reasoning non-classically (specifically, with geometric logic) you can eliminate the need for explicit topology: if you define points by a geometric theory, then the topology is implicit if you define a map by geometric constructions, then continuity is automatic
From this point of view, the purpose of explicit topology is to correct the errors introduced by classical reasoning.
We thus see sophisticated reasoning (classical principles) necessitating complicated thinking (topology), the reverse of what was intended. You don't need dogma to see this could be a bad idea, though you do need hard work to see whether the classical principles really can be dispensed with. Steve Vickers.