===== Neighbourhood =====
==== Set ====
| @#55CCEE: context     | @#55CCEE: $\langle X,\mathcal{T}_X\rangle$ ... topological space |
| @#55CCEE: context     | @#55CCEE: $p\in X$ |
| @#FFBB00: definiendum | @#FFBB00: $ U_p\in\mathrm{it} $ |
| @#55EE55: postulate   | @#55EE55: $ \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 $\mathbb R^n$:

| @#EEEE55: predicate   | @#EEEE55: $X$ ... 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:

| @#EEEE55: predicate   | @#EEEE55: $X$ ... topoloical manifold $ \equiv X$ ... Hausdorff space, locally euclidean space |

=== Reference ===
Wikipedia: 
[[http://en.wikipedia.org/wiki/Hausdorff_space|Hausdorff space]],
[[http://en.wikipedia.org/wiki/Topological_manifold|Topological manifold]]
==== Parents ====
=== Context ===
[[Topological space]]
=== Related ===
[[Homeomorphism]]