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Register a new userCardinal invariants for $kappa$-box products
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Definition. Let $kappa$ be an infinite cardinal, let {X(i)} be a (not necessarily faithfully indexed) set of topological spaces, and let X be the product of the spaces X(i). The $kappa$-box product topology on X is the topology generated by those products of sets U(i) for which (a) for each i, U(i) is open in X(i); and (b) U(i) = X(i) with fewer than $kappa$-many exceptions. (Thus, the usual Tychonoff product topology on X is the $omega$-box topology.) With emphasis on weight, density character, and Souslin number, the authors study and determine the value of several cardinal invariants on the space X with its $kappa$-box topology, in terms of the corresponding invariants of the individual spaces X(i). To the authors knowledge, this work is the first systematic study of its kind. Some of the results are axiom-sensitive, and some duplicate (and extend, and make precise) earlier work of Hewitt-Marczewski-Pondiczery, of Englking-Karlowicz, of Comfort-Negrepontis, and of Cater-Erdos-Galvin.
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By the {em Suslinian number} $Sln(X)$ of a continuum $X$ we understand the smallest cardinal number $kappa$ such that $X$ contains no disjoint family $C$ of non-degenerate subcontinua of size $|C|gekappa$. For a compact space $X$, $Sln(X)$ is the smallest Suslinian number of a continuum which contains a homeomorphic copy of $X$. Our principal result asserts that each compact space $X$ has weight $leSln(X)^+$ and is the limit of an inverse well-ordered spectrum of length $le Sln(X)^+$, consisting of compacta with weight $leSln(X)$ and monotone bonding maps. Moreover, $w(X)leSln(X)$ if no $Sln(X)^+$-Suslin tree exists. This implies that under the Suslin Hypothesis all Suslinian continua are metrizable, which answers a question of cite{DNTTT1}. On the other hand, the negation of the Suslin Hypothesis is equivalent to the existence of a hereditarily separable non-metrizable Suslinian continuum. If $X$ is a continuum with $Sln(X)<2^{aleph_0}$, then $X$ is 1-dimensional, has rim-weight $leSln(X)$ and weight $w(X)geSln(X)$. Our main tool is the inequality $w(X)leSln(X)cdot w(f(X))$ holding for any light map $f:Xto Y$.
A function $f:Xto Y$ between topological spaces is called $sigma$-$continuous$ (resp. $barsigma$-$continuous$) if there exists a (closed) cover ${X_n}_{ninomega}$ of $X$ such that for every $ninomega$ the restriction $f{restriction}X_n$ is continuous. By $mathfrak c_sigma$ (resp. $mathfrak c_{barsigma}$) we denote the largest cardinal $kappalemathfrak c$ such that every function $f:Xtomathbb R$ defined on a subset $Xsubsetmathbb R$ of cardinality $|X|<kappa$ is $sigma$-continuous (resp. $barsigma$-continuous). It is clear that $omega_1lemathfrak c_{barsigma}lemathfrak c_sigmalemathfrak c$. We prove that $mathfrak plemathfrak q_0=mathfrak c_{barsigma}=min{mathfrak c_sigma,mathfrak b,mathfrak q}lemathfrak c_sigmalemin{mathrm{non}(mathcal M),mathrm{non}(mathcal N)}$. The equality $mathfrak c_{barsigma}=mathfrak q_0$ resolves a problem from the initial version of the paper.
A function f from reals to reals (f:R->R) is almost continuous (in the sense of Stallings) iff every open set in the plane which contains the graph of f contains the graph of a continuous function. Natkaniec showed that for any family F of continuum many real functions there exists g:R->R such that f+g is almost continuous for every f in F. Let AA be the smallest cardinality of a family F of real functions for which there is no g:R->R with the property that f+g is almost continuous for every f in F. Thus Natkaniec showed that AA is strictly greater than the continuum. He asked if anything more could be said. We show that the cofinality of AA is greater than the continuum, c. Moreover, we show that it is pretty much all that can be said about AA in ZFC, by showing that AA can be equal to any regular cardinal between c^+ and 2^c (with 2^c arbitrarily large). We also show that AA = AD where AD is defined similarly to AA but for the class of Darboux functions. This solves another problem of Maliszewski and Natkaniec.
We study products of general topological spaces with Mengers covering property, and its refinements based on filters and semifilters. To this end, we extend the projection method from the classic real line topology to the Michael topology. Among other results, we prove that, assuming CH{}, every productively Lindelof space is productively Menger, and every productively Menger space is productively Hurewicz. None of these implications is reversible.
Given a collection of pairwise co-prime integers $% m_{1},ldots ,m_{r}$, greater than $1$, we consider the product $Sigma =Sigma _{m_{1}}times cdots times Sigma _{m_{r}}$, where each $Sigma _{m_{i}}$ is the $m_{i}$-adic solenoid. Answering a question of D. P. Bellamy and J. M. L ysko, in this paper we prove that if $M$ is a subcontinuum of $Sigma $ such that the projections of $M$ on each $Sigma _{m_{i}}$ are onto, then for each open subset $U$ in $Sigma $ with $Msubset U$, there exists an open connected subset $V$ of $Sigma $ such that $Msubset Vsubset U$; i.e. any such $M$ is ample in the sense of Prajs and Whittington [10]. This contrasts with the property of Cartesian squares of fixed solenoids $Sigma _{m_{i}}times Sigma _{m_{i}}$, whose diagonals are never ample [1].
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