In this note, we prove that an arbitrary product of $W$-spaces is a $\kappa$-Fr\'{e}chet-Urysohn space. We also show that the boundary tightness of a product $X \times Y$ is at most $\kappa$ provided that $X$ is compact with tightness $t(X) \leq \kappa$ and $Y$ has boundary tightness $tb(Y) \leq \kappa$. The first result fully resolves, and the second partially addresses, two questions recently raised by Tkachuk.
Let $A$ and $X$ be connected based CW complexes with $H_1(A;\mathbb Q)\neq 0$, and let $m\geq 1$ be the least degree with $\widetilde H_m(X;\mathbb Q)\neq0$. We prove that the Bousfield--Kan rational completion map of $A\vee X$ induces a map on $H_{2m}(-;\mathbb Q)$ whose cokernel has dimension at least $2^{\aleph_0}$....
In this paper we provide a positive version of Miljutin's theorem. Namely, we show that for any uncountable compact metric spaces $K$ and $L$ there exists a positive isomorphism $T:C(K)\longrightarrow C(L)$. Moreover, the inverse of $T$ is a regular operator, and $T$ may be chosen so that $\|T\|_\mathrm{r} \cdot \|T^{-...
Javier Laguna-Ricart, Gonzalo Martínez-Cervantes, Jakub Rondoš et al.· 0 citations
The Erd\H{o}s--Dushnik--Miller theorem states that for every aleph $\kappa$, \[ \kappa\to(\kappa,\omega); \] that is, every coloring $c:[\kappa]^2\to2$ has either a $0$-homogeneous set of cardinality $\kappa$ or a $1$-homogeneous set of cardinality $\omega$. In this article, we present a purely combinatorial proof of t...
Fix $k,s\in\mathbb{N}$. We prove that there exists a subadditive density on $\mathbb{N}$ such that, for every polynomial $P\in\mathbb{Z}[y]$ of degree $k$ and with $P(0)=0$, every set of positive density contains configurations $\{x,x+P(y),xy^s\}$ for arbitrarily large $x>y\geq 2$. This provides a density strengthening...
In this paper it is proved that there exists (in $ZFC$) a separable pseudocompact weak $\lambda$-space $X$ which is not $\Delta_1$-space. It follows that there exists a space $X$ such that $B_1(X,[0, 1])$ is Choquet, while $B_1(X)$ is not Choquet. Also it is proved that there exists a $\gamma$-set $X$ which is not weak...
In this paper, we study the rational realization of the gap-two product $K(\Q,n)\times K(\Q,n+2)$ as the classifying space $\B(X)$ for a simply-connected $\pi$-finite space $X$ with rational homology of finite type. We prove that $K(\Q,n)\times K(\Q,n+2)$ can be realized as $\B(X)$ up to rational homotopy equivalence i...
Yang Bai, Xiu-Gui Liu, Jia-Xi Zha· 0 citations
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