Does there exist a generalization of a manifold whereby instead of being locally $\mathbb{R}^n$, it's locally another specified space?
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1$\begingroup$ Yes - the right term to use is that of a ringed space. But this question is better suited for math.stackexchange $\endgroup$– Georg LehnerCommented Mar 10, 2016 at 16:01
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2$\begingroup$ Could you specify which kind of local model you are thinking about? Topological spaces with particular properties (e.g., Cantor sets), algebraically defined spaces, or maybe Banach spaces? The way the question is currently phrased, it looks too broad for a clear answer. @GeorgLehner I don't think that ringed spaces are the only or the obvious solution here. Typically, you get a huge class of local models (e.g., all affine varieties if you are doing algebraic geometry) instead of "one specified space". Or, you need a severe restriction on the admissible rings. $\endgroup$– Sebastian GoetteCommented Mar 10, 2016 at 16:32
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1$\begingroup$ See the nearly identically worded mathematics stackexchange question does there exist a generalization of manifold, which was asked 15 hours before this math overflow question was asked. $\endgroup$– Dave L RenfroCommented Mar 10, 2016 at 17:36
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$\begingroup$ @DaveLRenfro what can i say, you got me $\endgroup$– dingakur prejamangokurCommented Mar 10, 2016 at 17:42
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$\begingroup$ @SebastianGoette : What I meant was that ringed spaces (or more generally ringed toposes) are just the right generalization to make sense of the phrase "being locally isomorphic to". Any fixed locally ringed space gives you a notion of "manifolds" modeled on that space. $\endgroup$– Georg LehnerCommented Mar 10, 2016 at 17:55
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