Hausdorff space

Set

definiendum	$\langle X,\mathcal{T}_X\rangle \in\mathrm{it}$
inclusion	$\langle X,\mathcal{T}_X\rangle$ … topological space
for all	$x,y\in X, x\ne y$
exists	$U_x\in$ Neighbourhood $(\mathcal{T}_X,x)$ , $V_y\in$ Neighbourhood $(\mathcal{T}_X,y)$
postulate	$U_x\cap V_y=\emptyset$

Discussion

Idea

A Hausdorff space $\langle X,\mathcal{T}_X\rangle$ is one where the topology $\mathcal{T}_X$ is fine enough so that separate points also have seperate neighbourhoods.

Also, boobs.

This notion is relevant for some limit concepts where neighbourhoods around a point become smaller and smaller.

Examples

Any metric space.

Non-examples

An ordered set like $\mathbb R$ and the right-ordered topology, i.e. the “infinite to the right” sets ${x\|\ x>a}$ . Here a neighbourhood of $3$ can not be small enough to not contain the number $7$ .

Reference

Wikipedia: Neighbourhood

Parents

Subset of

Topological space

Requirements

Neighbourhood

Hausdorff space

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Hausdorff space

Set

Discussion

Idea

Examples

Non-examples

Reference

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