So in my wikipedia readings, I have often stumbled across mentions of topological manifolds. I thought a manifold intrinsically had to be topological, either explicitly or implicitly like in a metric space. I don't even know how to conceptualise a non-topological manifold. Is this just wikipedia being weird, or are non-toplogical manifolds a thing, and if they are, what are they? Are there any examples I can see? And do they generalise topological manifolds, or are they their own thing?
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$\begingroup$ I see no reason why manifolds couldn't be based on Cech closure spaces or other generalized topological structures, although I doubt much of interest would arise from such things (by the way, saying this guarantees the opposite). However, the answer by @Arnaud Mortier is the real reason. $\endgroup$– Dave L. RenfroAug 11, 2018 at 11:27
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You use topological to emphasize the fact that your manifold doesn't have additional structure or properties, such as a differentiable structure, or a piecewise linear structure, where the transition maps have extra requirements on top of being continuous.
If you were to write "let $X$ be a manifold", the reader would possibly wonder what class of manifolds you are considering.
answered Aug 11, 2018 at 9:47