I think everything said here can applies to just topologies.

 That isn’t really true. Dimension isn’t so well defined in topology but is reasonably straightforward with a manifold. The article also touched on geometry (sum of angles of a triangle). To be topological a bunch of things must change. The bug walking example doesn’t work and is demoted to an analogy. All notion of distance and direction is lost. If the article were about topology then it would probably be talking about distinguishing shapes by homotopy/homology rather than lines and walks and angles.The article also gets some details a bit wrong/fuzzy: it says you can tell if you are on a sphere by walking in a straight line infinitely far and seeing if you ever cross yourself. But this property also follows on the surface of a (rounded at the top) cone. Even if you require this property in all directions you get problems on eg a torus.
 If I'm not mistaken, dimension can be defined easily for a topological manifold, which is actually a more fundamental structure than a differential manifold. (The former requires that chart overlaps be homeomorphisms, i.e. continuous bijections), while the latter requires that they be diffeomorphisms, i.e. smooth bijections). Smooth manifolds don't form a nice category for technical reasons, but one can think of a forgetful functor from differential manifolds to topological manifolds, and dimension being defined in the latter.
 I don’t disagree that one can define dimension for a manifold (topological or differentiable). I was replying to the parent comment and so I was pointing out that dimension isn’t really well defined for a general topological space.
 Thanks for pointing out the fuzziness! I did not want to extend this further so as not to confuse readers who have no familiarity with the subject. Of course, the fact that you cross your own path at some point is _not_ indicative of living on a sphere; there are numerous other manifolds that satisfy this (not to speak of the torus, as you yourself mentioned).I will try to be more precise about this in a subsequent article!
 Don't. Precision obfuscates more than it increases resolution.You don't want to be precise. You want to increase the resolution of the key ideas being seen. This is also why parables are so popular for teaching: they're literally not true, but they resolve to an image of something that is true. Please don't introduce homotopy/homology. Lines and angles are perfect.

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