Let L be an Archimedean Riesz space with a weak order unit u. A sufficient condition under which Dedekind [σ-]completeness of the principal ideal Au can be lifted to L is given (Lemma). This yields a concise proof of two theorems of Luxemburg and Zaanen concerning projection properties of C(X)-spaces. Similar results are obtained for the Riesz spaces Bn(T ), n = 1, 2, . . . , of all functions o...

In this paper we further a previous study concerning abstract regularity for monotone set multifunctions, with has immediate applications in well-known situations such as the Borel σ-algebra of a Hausdorff space and/or the Borel (Baire, respectively) δ-ring or σ-ring of a locally compact Hausdorff space. We also study relationships among abstract regularities and other properties of continuity....

We prove that the class of functions with the Baire property has the weak difference property in category sense. That is, every function for which f(x+h)−f(x) has the Baire property for every h ∈ R can be written in the form f = g+H+φ where g has the Baire property, H is additive, and for every h ∈ R we have φ(x+h)−φ(x) 6= 0 only on a meager set. We also discuss the weak difference property of ...

For any non-totally disconnected Polish space, there is a family of c = 2א0 many Wadge incomparable finite level Borel subsets. If the space is additionally locally compact or locally connected, there is a family of 2 many Wadge incomparable subsets. In this note, we study the Wadge order for Polish spaces which are not totally disconnected. For a fixed Polish space, the Wadge order for the spa...

In this paper, will be presented the concept of Borel sets and Baire on normal topological spaces locally compact as new types λ – system study basic characteristics related to each them obtain results them.

We deene the continuum up to order isomorphism (and hence homeomorphism) as the nal coalgebra of the functor X !, ordinal product with !. This makes an attractive analogy with the deenition of the ordinal ! itself as the initial algebra of the functor 1; X, prepend unity, with both deenitions made in the category of posets. The variants 1; (X !), X o !, and 1; (X o !) yield respectively Cantor ...

We show that the Baire measurable chromatic number of every locally finite Borel graph on a non-empty Polish space is strictly less than twice its ordinary chromatic number, provided this ordinary chromatic number is finite. In the special case that the connectedness equivalence relation is hyperfinite, we obtain the analogous result for the μ-measurable chromatic number.

Almost all representations considered in computable analysis are partial. We provide arguments in favor of total representations (by elements of the Baire space). Total representations make the well known analogy between numberings and representations closer, unify some terminology, simplify some technical details, suggest interesting open questions and new invariants of topological spaces rele...