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Let X and K be a Cech-complete topological group and a compact group, respectively. We prove that if G is a non-equicontinuous subset of C H o m ( X , K ) , the set of all continuous homomorphisms of X into K, then there is a countably infinite subset L ⊆ G such that L ‾ K X is canonically homeomorphic to βω, the Stone–Cech compactifcation of the natural numbers. As a consequence, if G is an infinite subset of C H o m ( X , K ) such that for every countable subset L ⊆ G and compact separable subset Y ⊆ X it holds that either L ‾ K Y has countable tightness or | L ‾ K Y | ≤ c , then G is equicontinuous. Given a topological group G, denote by G + the (algebraic) group G equipped with the Bohr topology. It is said that G respects a topological property P when G and G + have the same subsets satisfying P . As an application of our main result, we prove that if G is an abelian, locally quasiconvex, locally k ω group, then the following holds: (i) G respects any compact-like property P stronger than or equal to functional boundedness; (ii) G strongly respects compactness.

Objective Recently, the DSM-5 has developed a new diagnostic category named "Substance-related and Addictive Disorders". This category includes gambling disorder (GD) as the sole behavioral addiction, but does not include sex addiction (SA). The aim of this study is to investigate whether SA should be classified more closely to other behavioral addictions, via a comparison of the personality characteristics and comorbid psychopathology of individuals with SA with those of individuals with GD, which comes under the category of addiction and related disorders. Method The sample included 59 patients diagnosed with SA, who were compared to 2190 individuals diagnosed with GD and to 93 healthy controls. Assessment measures included the Diagnostic Questionnaire for Pathological Gambling, the South Oaks Gambling Screen, the Symptom CheckList-90 Items-Revised and the Temperament and Character Inventory-Revised. Results No statistically significant differences were found between the two clinical groups, except for socio-economic status. Although statistically significant differences were found between both clinical groups and controls for all scales on the SCL-90, no differences were found between the two clinical groups. The results were different for personality characteristics: logistic regression models showed that sex addictive behavior was predicted by a higher education level and by lower scores for TCI-R novelty-seeking, harm avoidance, persistence and self-transcendence. Being employed and lower scores in cooperativeness also tended to predict the presence of sex addiction. Conclusions While SA and GD share some psychopathological and personality traits that are not present in healthy controls, there are also some diagnostic-specific characteristics that differentiate between the two clinical groups. These findings may help to increase our knowledge of phenotypes existing in behavioral addictions.

Combining ideas of Troallic and Cascales, Namioka, and Vera, we prove several characterizations of \textit{almost equicontinuity} and \textit{hereditary almost equicontinuity} for subsets of metric-valued continuous functions when they are defined on a \v{C}ech-complete space. We also obtain some applications of these results to topological groups and dynamical systems., Comment: 27 pages

The notion of locally quasi-convex abelian group, introduced by Vilenkin, is extended to maximally almost periodic non-necessarily abelian groups. For that purpose, we look at certain bornologies that can be defined on the set rep(G)rep(G) of all finite dimensional continuous representations on a topological group G in order to associate well behaved group topologies (dual topologies) to them. As a consequence, the poset of all Hausdorff totally bounded group topologies on a group G is shown to be isomorphic to the poset of certain special subsets of rep(Gd)rep(Gd). Moreover, generalizing some ideas of Namioka, we relate the structural properties of the dual topological groups to topological properties of the bounded subsets belonging to the associate bornology. In like manner, certain type of bornologies that can be defined on a group G allow one to define canonically associate uniformities on the dual object View the MathML sourceGˆ. As an application, we prove that if for every dense subgroup H of a compact group G we have that if View the MathML sourceHˆ and View the MathML sourceGˆ are uniformly isomorphic, then G is metrizable. Thereby, we extend to non-abelian groups some results previously considered for abelian topological groups.