Basic Notions of Manifold
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.
The point `x\in M`, M is a manifold. U is a neighborhood of x, and
there is a homeomorphism `\phi:U\mapsto \phi(U)`. then, Any such a
$(U,\phi(U))$ means a realization of any point of U: $\phi(y)$ is