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Register a new userCardinal invariants concerning functions whose sum is almost continuous
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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.
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We construct, under the assumption that union of less than continuum many meager subsets of R is meager in R, an additive connectivity function f:R-->R with Cantor intermediate value property which is not almost continuous. This gives a partial answer to a question of D. Banaszewski. We also show that every extendable function g:R-->R with a dense graph satisfies the following stronger version of the SCIVP property: for every a<b and every perfect set K between g(a) and g(b) there is a perfect subset C of (a,b) such that g[C] subset K and g|C is continuous strictly increasing. This property is used to construct a ZFC example of an additive almost continuous function f:R-->R which has the strong Cantor intermediate value property but is not extendable. This answers a question of H. Rosen. This also generalizes Rosens result that a similar (but not additive) function exists under the assumption of the continuum hypothesis.
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.
We continue the study of the virtual large cardinal hierarchy, initiated in Gitman and Schindler (2018), by analysing virtu
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.
In [5], Hjorth proved that for every countable ordinal $alpha$, there exists a complete $mathcal{L}_{omega_1,omega}$-sentence $phi_alpha$ that has models of all cardinalities less than or equal to $aleph_alpha$, but no models of cardinality $aleph_{alpha+1}$. Unfortunately, his solution does not yield a single $mathcal{L}_{omega_1,omega}$-sentence $phi_alpha$, but a set of $mathcal{L}_{omega_1,omega}$-sentences, one of which is guaranteed to work. It was conjectured in [9] that it is independent of the axioms of ZFC which of these sentences has the desired property. In the present paper, we prove that this conjecture is true. More specifically, we isolate a diagonalization principle for functions from $omega_1$ to $omega_1$ which is a consequence of the Bounded Proper Forcing Axiom (BPFA) and then we use this principle to prove that Hjorths solution to characterizing $aleph_2$ in models of BPFA is different than in models of CH. In addition, we show that large cardinals are not needed to obtain this independence result by proving that our diagonalization principle can be forced over models of CH.
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