| 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 $ |
A neighbourhood of $p$ is a reasonably big set surrounding $p$.
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 |
Wikipedia: Hausdorff space, Topological manifold