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We say that a cardinal function $φ$ reflects an infinite cardinal $κ$ , if given a topological space $X$ with $φ (X) \geq κ$ , there exists $Y \in {[X]}^{\leq κ}$ with $φ (Y) \geq κ$ . We investigate some problems, discussed by Hodel and Vaughan in Reflection theorems for cardinal functions, Topology Appl. 100 (2000), 47–66, and Juhász in Cardinal functions and reflection, Topology Atlas Preprint no. 445, 2000, related to the reflection for the cardinal functions character and pseudocharacter. Among other results, we present some new equivalences with...

Reflecting Lindelöf and converging ω₁-sequences

Alan Dow, Klaas Pieter Hart (2014)

Fundamenta Mathematicae

We deal with a conjectured dichotomy for compact Hausdorff spaces: each such space contains a non-trivial converging ω-sequence or a non-trivial converging ω₁-sequence. We establish that this dichotomy holds in a variety of models; these include the Cohen models, the random real models and any model obtained from a model of CH by an iteration of property K posets. In fact in these models every compact Hausdorff space without non-trivial converging ω₁-sequences is first-countable and, in addition,...

Reflecting topological properties in continuous images

Vladimir Tkachuk (2012)

Open Mathematics

Given a topological property P, we study when it reflects in small continuous images, i.e., when for some infinite cardinal κ, a space X has P if and only if all its continuous images of weight less or equal to κ have P. We say that a cardinal invariant η reflects in continuous images of weight κ + if η(X) ≤ κ provided that η(Y) ≤ κ whenever Y is a continuous image of X of weight less or equal to κ +. We establish that, for any infinite cardinal κ, the spread, character, pseudocharacter and Souslin...

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$R_{0}$ and $R_{1}$ axioms in fuzzy topology

$R$ -separated spaces.

Ramsey Theorems and the Property of Baire

Rapid ultrafilter need not be Q-point

Rapport entre l'ensemble des ultrafiltres admettant un ultrafiltre donné pour image et l'ensemble des images de cet ultrafiltre

Raster convergence with respect to a closure operator

Rationals as a non-trivial complete convergence group

Rationals with exotic convergences

Real closed rings, I. Residue rings of rings of continuous functions

Real ideals in the maximal ring of quotients of $C (X)$

Realisierung und Auswahlaxiom

Realizations of topologies and closure operators by set systems and by neighbourhoods

Refining Families for Ultrafilters.

Reflecting character and pseudocharacter

Reflecting Lindelöf and converging ω₁-sequences

Reflecting topological properties in continuous images

Reflective functors via nearness

Regular and normal quantales

Regular Convergence Spaces.

Regular fuzzy spaces and fuzzy almost regular spaces

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