- #1

dumbQuestion

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So to be a manifold, each point in the space should have a neighborhood about it that is homeomorphic to R^n. (and n will tell you what dimension the manifold is). To start, what is 0-dimension Euclidean space? Would this be just the empty set?

Now secondly I start to think about what a homeomorphism implies. It's a bijection between two topological spaces where each direction of the mapping is continuous. (so open sets in one map to open sets in the other). This is what makes it difficult for me to understand how a discrete space could be a manifold - how would you get a bijection between a discrete space and euclidean space? I can understand it in one direction by doing something trivial (like say, every set in the discrete space S maps to the empty set or all of R or something like that). But how do you get it in the other direction?