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When we mention Euclid space, maybe it induct excess meanings.
So, we can try to construct an object that its meaning is less than Euclid space but however, is meaningful and analyzable.
At least, we need some points, then, if we want a point is analyzable, it must has its neighborhood, moreover, to make it computable or realizable, let its neighborhood homeomorphism with an open set of Euclid space.
Such points constitutes a set, then make this set is a separated set, a Hausdorff set, we definite such a set a manifold.