On the separable quotient problem for Banach spaces

While the classic separable quotient problem remains open, we survey general results related to this problem and examine the existence of a particular infinitedimensional separable quotient in some Banach spaces of vector-valued functions, linear operators and vector measures. Most of the results presented are consequence of known facts, some of them relative to the presence of complemented copies of the classic sequence spaces c_0 and l_p, for 1 <= p <= \infty. Also recent results of Argyros - Dodos - Kanellopoulos, and Sliwa are provided. This makes our presentation supplementary to a previous survey (1997) due to Mujica

Covering Properties of Cp(X) and Ck(X)

[EN] Let X be a Tychonoff space. We survey some classic and recent results that characterize the topology or cardinality of X when C-p (X) or C-k (X) is covered by certain families of sets (sequences, resolutions, closure-preserving coverings, compact coverings ordered by a second countable space) which swallow or not some classes of sets (compact sets, functionally bounded sets, pointwise bounded sets) in C(X).Research supported by Grant PGC2018-094431-B-I00 of Ministry of Science, Innovation & Universities of Spain.Ferrando, JC.; López Pellicer, M. (2020). Covering Properties of Cp(X) and Ck(X). Filomat. 34(11):3575-3599. https://doi.org/10.2298/FIL2011575FS35753599341

Spaces C(X)with ordered bases

[EN] The concept of Sigma-base of neighborhoods of the identity of a topological group G is introduced. If the index set Sigma subset of N-N is unbounded and directed (and if additionally each subset of Sigma which is bounded in N-N has a bound at Sigma) a base {U-alpha : alpha is an element of Sigma} of neighborhoods of the identity of a topological group G with U-beta subset of U-alpha whenever alpha <= beta with alpha, beta is an element of Sigma is called a Sigma-base (a Sigma(2)-base). The case when Sigma = N-N has been noticed for topological vector spaces (under the name of G-base) at [2]. If X is a separable and metrizable space which is not Polish, the space C-c(X) has a Sigma-base but does not admit any G-base. A topological group which is Frechet-Urysohn is metrizable iff it has a Sigma(2)-base of the identity. Under an appropriate ZFC model the space C-c (omega(1)) has a Sigma(2)-base which is not a G-base. We also prove that (i) every compact set in a topological group with a Sigma(2)-base of neighborhoods of the identity is metrizable, (ii) a C-p (X) space has a Sigma(2)-base iff X is countable, and (iii) if a space C-c(X) has a Sigma(2)-base then X is a C-Suslin space, hence C-c(X)is angelic. (C) 2016 Elsevier B.V. All rights reserved.Supported by Grant PROMETEO/2013/058 of the Conselleria de Education, Investigacion, Cultura y Deportes of Generalitat Valenciana. The second author also supported by the GACR Project 16-34860L and RVO: 67985840Ferrando, JC.; Kakol, J.; López Pellicer, M. (2016). Spaces C(X)with ordered bases. Topology and its Applications. 208:30-39. https://doi.org/10.1016/j.topol.2016.05.006S303920

Compact covers and function spaces

For a Tychonoff space X, we denote by Cp(X) and Cc(X) the space of continuous real-valued functions on X equipped with the topology of pointwise convergence and the compact-open topology respectively. Providing a characterization of the Lindelof $Sigma$-property of X in terms of Cp(X), we extend Okunev's results by showing that if there exists a surjection from Cp(X) onto Cp(Y) (resp. from Lp(X) onto Lp(Y)) that takes bounded sequences to bounded sequences, then $nu$Y is a Lindelof $Sigma$-space (respectively K-analytic) if $nu$X has this property. In the second part, applying Christensen's theorem, we extend Pelant's result by proving that if X is a separable completely metrizable space and Y is first countable, and there is a quotient linear map from Cc(X) onto Cc(Y), then Y is a separable completely metrizable space.We study also a non-separable case, and consider a different approach to the result of J. Baars, J. de Groot, J. Pelant and V. Valov, which is based on the combination of two facts: Complete metrizability is preserved by lp-equivalence in the class of metric spaces (J. Baars, J. de Groot, J. Pelant). If X is completely metrizable and lp-equivalent to a first-countable Y, then Y is metrizable (V. Valov). Some additional results are presented.The research was supported for the first named author by National Center of Science, Poland, Grant No. N N201 605340, and for the first and second named authors by Generalitat Valenciana, Conselleria d'Educacio i Esport, Spain, Grant PROMETEO/2013/058.Kakol, J.; López Pellicer, M.; Okunev, O. (2014). Compact covers and function spaces. Journal of Mathematical Analysis and Applications. 411(1):372-380. https://doi.org/10.1016/j.jmaa.2013.09.046S372380411

Intensity of physical education classes in adolescents

Se registró la frecuencia cardiaca de 182 estudiantes (97 chicos y 85 chicas) de entre 12 y 18 años durante sus clases de Educación Física. Los resultados muestran una media del 21,62±14,33% del tiempo de clase en valores MVPA (moderate to vigorous physical activity). Respecto al género, pese a no ser significativo, los mayores valores corresponden a la chicas (23,47±14,45% vs 19,99±14,10%; p=0,106). No se ha observado efecto del tipo de sesión (deportes colectivos, deportes individuales, juegos tradicionales o bailes) sobre el tiempo en valores MVPA (p>0,05; TE0.05; ES<0.020), obtaining the highest values in team sports sessions. Results show that intensity and duration of analyzed classes do not comply with recommendations to become an adequate cardiovascular exercis