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Egoroff, σ, and convergence properties in some archimedean vector lattices

A. W. Hager, J. van Mill (2015)

Studia Mathematica

An archimedean vector lattice A might have the following properties: (1) the sigma property (σ): For each a ₙ n ∈ ℕ c o n A ⁺ there are λ ₙ n ∈ ℕ ⊆ ( 0 , ∞ ) and a ∈ A with λₙaₙ ≤ a for each n; (2) order convergence and relative uniform convergence are equivalent, denoted (OC ⇒ RUC): if aₙ ↓ 0 then aₙ → 0 r.u. The conjunction of these two is called strongly Egoroff. We consider vector lattices of the form D(X) (all extended real continuous functions on the compact space X) showing that (σ) and (OC ⇒ RUC) are equivalent, and equivalent...

Essential P -spaces: a generalization of door spaces

Emad Abu Osba, Melvin Henriksen (2004)

Commentationes Mathematicae Universitatis Carolinae

An element f of a commutative ring A with identity element is called a von Neumann regular element if there is a g in A such that f 2 g = f . A point p of a (Tychonoff) space X is called a P -point if each f in the ring C ( X ) of continuous real-valued functions is constant on a neighborhood of p . It is well-known that the ring C ( X ) is von Neumann regular ring iff each of its elements is a von Neumann regular element; in which case X is called a P -space. If all but at most one point of X is a P -point, then X is called...

Exponential domination in function spaces

Vladimir Vladimirovich Tkachuk (2020)

Commentationes Mathematicae Universitatis Carolinae

Given a Tychonoff space X and an infinite cardinal κ , we prove that exponential κ -domination in X is equivalent to exponential κ -cofinality of C p ( X ) . On the other hand, exponential κ -cofinality of X is equivalent to exponential κ -domination in C p ( X ) . We show that every exponentially κ -cofinal space X has a κ + -small diagonal; besides, if X is κ -stable, then n w ( X ) ≤ κ . In particular, any compact exponentially κ -cofinal space has weight not exceeding κ . We also establish that any exponentially κ -cofinal space X with...

Exponential separability is preserved by some products

Vladimir Vladimirovich Tkachuk (2022)

Commentationes Mathematicae Universitatis Carolinae

We show that exponential separability is an inverse invariant of closed maps with countably compact exponentially separable fibers. This implies that it is preserved by products with a scattered compact factor and in the products of sequential countably compact spaces. We also provide an example of a σ -compact crowded space in which all countable subspaces are scattered. If X is a Lindelöf space and every Y ⊂ X with | Y | ≤ 2 ω 1 is scattered, then X is functionally countable; if every Y ⊂ X with | Y | ≤ 2 𝔠 is scattered, then...

Extending real-valued functions in βκ

Alan Dow (1997)

Fundamenta Mathematicae

An Open Coloring Axiom type principle is formulated for uncountable cardinals and is shown to be a consequence of the Proper Forcing Axiom. Several applications are found. We also study dense C*-embedded subspaces of ω*, showing that there can be such sets of cardinality c and that it is consistent that ω*{pis C*-embedded for some but not all p ∈ ω*.

Extending the ideal of nowhere dense subsets of rationals to a P-ideal

Rafał Filipów, Nikodem Mrożek, Ireneusz Recław, Piotr Szuca (2013)

Commentationes Mathematicae Universitatis Carolinae

We show that the ideal of nowhere dense subsets of rationals cannot be extended to an analytic P-ideal, F σ ideal nor maximal P-ideal. We also consider a problem of extendability to a non-meager P-ideals (in particular, to maximal P-ideals).

Extension of multisequences and countably uniradial classes of topologies

Szymon Dolecki, Andrzej Starosolski, Stephen W. Watson (2003)

Commentationes Mathematicae Universitatis Carolinae

It is proved that every non trivial continuous map between the sets of extremal elements of monotone sequential cascades can be continuously extended to some subcascades. This implies a result of Franklin and Rajagopalan that an Arens space cannot be continuously non trivially mapped to an Arens space of higher rank. As an application, it is proved that if for a filter ℋ on ω , the class of ℋ -radial topologies contains each sequential topology, then it includes the class of subsequential topologies....

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