regular space

(redirected from Regular Hausdorff space)

regular space

[¦reg·yə·lər ′spās]
(mathematics)
A topological space such that any neighborhood of a point in the space contains the closure of another neighborhood of the same point.
References in periodicals archive ?
P.Nanzetta and D Plank [7] extended some characterization to an arbitrary completely regular Hausdorff space X and derived some corollaries.
In the sequel, unless otherwise stated, X is a nonempty completely regular Hausdorff space. We represent by [C.sub.p](X) the ring C(X) of real-valued continuous functions defined on X equipped with the pointwise topology [[tau].sub.p].
([1, Theorem IV.9.4]) If the realcompactification vX of a completely regular Hausdorff space X is a Lindelof [SIGMA]-space, then there exists a Lindelof [SIGMA]-space Z such that [C.sub.p] (X) [subset or equal to] Z [subset or equal to] [R.sup.X].
Now assume that [OMEGA] is a completely regular Hausdorff space. Let Ba denote the [sigma]-algebra of Baire sets in [OMEGA], which is the [sigma]-algebra generated by the class Z of all zero sets of bounded continuous positive functions on [omega].
Pytkeev [13] and independently Gerlits [7], see also [1] and [10], proved that, for a completely regular Hausdorff space X the space [C.sub.p](X) of continuous real-valued functions on X is Frechet-Urysohn if and only if [C.sub.p](X) is sequential iff [C.sub.p](X) is a k-space.