Regular topological space

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1 Definition
2 See also
3 References

Definition

A topological space, $(X,\mathcal{ J })$ is regular if^[1]:

$\forall E\in C(\mathcal{J})\ \forall x\in X-E\ \exists U,V\in\mathcal{J}[U\cap V=\emptyset\implies(E\subset U\wedge x\in V)]$ - (here $C(\mathcal{J})$ denotes the closed sets of the topology $\mathcal{J}$ )

Note that it is $E\subset U$ not $\subseteq$ , the author (^[1]) like me is pedantic about this, so it must matter

TODO: Investigate consequences/differences between $E\subseteq U$ and $E\subset U$

TODO: Picture

See also

References

↑ ^{Jump up to: 1.0} ^1.1 Introduction to Topology - Theodore W. Gamelin & Robert Everist Greene

v • d • e Topology

Overview of the structures studied in topology

Primitives	topological space, open set, neighbourhood

Global properties of topological spaces	Compactness, Connectedness (Path-Connectedness), Hausdorff-ness

Local properties of topological spaces

Constructs	product (see also box, difference between), subspace (AKA: induced), quotient, disjoint union, adjunction, cone over a topological space

Subtypes of topological spaces	inner product spaces $\subset$ normed spaces $\subset$ metric spaces $\subset$ topological spaces

Maps between topological spaces	Continuous map (AKA: topological homomorphism), topological homeomorphism

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