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Let us suppose $X$ is a topological manifold, which we define as a Ringed Space with the local model being the Euclidean space $\mathbb R^n$ with the sheaf of continuous functions on it.

Normally, for most definitions of various structures using the ringed space idea, it is required that the structure sheaf is locally ringed. However, in this case, because of the locally Euclidean property, does it not automatically follow?

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Do you even need the sheaf of rings? Isn't a topological manifold just a topological space locally homeomorphic to euclidean space? – Zhen Lin Feb 7 '13 at 8:14

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