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?
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.