In this paper we consider relational T-algebras, objects in (T,2)-Cat, as spaces and we explore the topological property of T-regularity. This notion goes back to Möbus [18] who introduced it in a more general abstract framework. When applied to the ultrafilter monad ▪ and to the well known lax-algebraic presentation of Top as (▪,2)-Cat, ▪-regularity is known to be equivalent to the usual regularity of the topological space [5]. We prove that in general for a power-enriched monad T with the Kleisli extension, even when restricting to proper elements, T-regularity is too strong since in most cases it implies the object being indiscrete.For the lax-algebraic presentations of Top as (F,2)-Cat, via the power-enriched filter monad F and of App as (I,2)-Cat, via the power-enriched functional ideal monad I, we present weaker conditions in terms of convergence of filters and functional ideals respectively, equivalent to the usual regularity in Top and App.For the lax-algebraic presentation of App as (B,2)-Cat, via the prime functional ideal monad B, a submonad of I with the initial extension to Rel, restricting to proper elements already gives more interesting results. We prove that B-regularity (restricted to proper prime functional ideals) is equivalent to the approach space being topological and regular. However it requires further weakening of the concept to obtain a characterization of the usual regularity in App in terms of convergence of prime functional ideals.