Neighbourhood

Set

context $\langle X,\mathcal{T}_X\rangle$ … topological space
context $p\in X$
definiendum $ U_p\in\mathrm{it} $
postulate $ \exists(\mathcal{O}\in\mathcal{T}_X).\ \mathcal{O}\subseteq U_p $

Discussion

A neighbourhood of $p$ is a reasonably big set surrounding $p$.

Predicates

Consider $X$ together with a topology, then

locally euclidean space means $X$ is homeomorphic to $\mathbb R^n$:

predicate $X$ … locally euclidean space $ \equiv \forall(x\in X).\ \exists(U_x\in\mathrm{Neighbourhood}(x)),\ f.\ f\in\mathrm{Homeomorphism}(U_x,\mathbb R^n)$

topoloical manifold means Hausdorff space + locally euclidean space:

predicate $X$ … topoloical manifold $ \equiv X$ … Hausdorff space, locally euclidean space

Reference

Wikipedia: Hausdorff space, Topological manifold

Parents

Context

Topological space

Homeomorphism