More on the cardinality of a topological space

<p>In this paper we continue to investigate the impact that various separation axioms and covering properties have onto the cardinality of topological spaces. Many authors have been working in that field. To mention a few, let us refer to results by Arhangel’skii, Alas, Hajnal-Juhász, Bell-Gisburg-Woods, Dissanayake-Willard, Schröder and to the excellent survey by Hodel “Arhangel’skii’s Solution to Alexandroff’s problem: A survey”.</p><p>Here we provide improvements and analogues of some of the results obtained by the above authors in the settings of more general separation axioms and cardinal invariants related to them. We also provide partial answer to Arhangel’skii’s question concerning whether the continuum is an upper bound for the cardinality of a Hausdorff Lindelöf space having countable pseudo-character (i.e., points are Gδ). Shelah in 1978 was the first to give a consistent negative answer to Arhangel’skii’s question; in 1993 Gorelic established an improved result; and further results were obtained by Tall in 1995. The question of whether or not there is a consistent bound on the cardinality of Hausdorff Lindelöf spaces with countable pseudo-character is still open. In this paper we introduce the Hausdorff point separating weight Hpw(X), and prove that (1) |X| ≤ Hpsw(X)^aL^c^(X)ψ(X), for Hausdorff spaces and (2) |X| ≤ Hpsw(X)^ω^L^c^(X)ψ(X), where X is a Hausdorff space with a π-base consisting of compact sets with non-empty interior. In 1993 Schröder proved an analogue of Hajnal and Juhasz inequality |X| ≤ 2^c(X)χ(X) for Hausdorff spaces, for Urysohn spaces by considering weaker invariant - Urysohn cellularity Uc(X) instead of cellularity c(X). We introduce the n-Urysohn cellularity n-Uc(X) (where n≥2) and prove that the previous inequality is true in the class of n-Urysohn spaces replacing Uc(X) by the weaker n-Uc(X). We also show that |X| ≤ 2^Uc(X)πχ(X) if X is a power homogeneous Urysohn space.</p>

Less Equal, Less Trusting? Longitudinal and Cross-sectional Effects of Income Inequality on Trust in U.S. States, 1973–2012

Does income inequality reduce social trust? Although both popular and scholarly accounts have argued that income inequality reduces trust, some recent research has been more skeptical, noting these claims are more robust cross-sectionally than longitudinally. Furthermore, although multiple mechanisms have been proposed for why inequality could affect trust, these have rarely been tested explicitly. I examine the effect of state-level income inequality on trust using the 1973–2012 General Social Surveys. I find little evidence that states that have been more unequal over time have less trusting people. There is some evidence that the growth in income inequality is linked with a decrease in trust, but these effects are sensitive to how time is accounted for. While much previous inequality and trust research has focused on status anxiety, this mechanism receives the little support, but mechanisms based on social fractionalization and on exploitation and resentment receive some support. This analysis improves on previous estimates of the effect of state-level inequality on trust by using far more available observations, accounting for more potential individual and state level confounders, and using higher-quality income inequality data based on annual IRS tax returns. It also contributes to our understanding of the mechanism(s) through which inequality may affect trust.

A Note on Generalized Hardy-Sobolev Inequalities

We are concerned with finding a class of weight functions g so that the following generalized Hardy-Sobolev inequality holds: ∫Ωgu2≤C∫Ω|∇u|2, u∈H01(Ω), for some C>0, where Ω is a bounded domain in ℝ2. By making use of Muckenhoupt condition for the one-dimensional weighted Hardy inequalities, we identify a rearrangement invariant Banach function space so that the previous integral inequality holds for all weight functions in it. For weights in a subspace of this space, we show that the best constant in the previous inequality is attained. Our method gives an alternate way of proving the Moser-Trudinger embedding and its refinement due to Hansson.

On a Result of Levin and Stečkin

The following inequality for0<p<1andan≥0originates from a study of Hardy, Littlewood, and Pólya:∑n=1∞((1/n)∑k=n∞ak)p≥cp∑n=1∞anp. Levin and Stečkin proved the previous inequality with the best constantcp=(p/(1-p))pfor0<p≤1/3. In this paper, we extend the result of Levin and Stečkin to0<p≤0.346.