Neighbourhood

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