Generalized independence Hernández-Hernández, Fernando; López-Callejas, Carlos
Annals of pure and applied logic,
July 2024, 2024-07-00, Volume:
175, Issue:
7
Journal Article
Peer reviewed
Open access
We explore different generalizations of the classical concept of independent families on ω following the study initiated by Kunen, Fischer, Eskew and Montoya. We show that under (Dℓ)κ⁎ we can get ...strongly κ-independent families of size 2κ and present an equivalence of GCH in terms of strongly independent families. We merge the two natural ways of generalizing independent families through a filter or an ideal and we focus on the C-independent families, where C is the club filter. Also we show a relationship between the existence of J-independent families and the saturation of the ideal J.
We give a combinatorial characterization of countable submaximal subspaces of 2κ. Using a parametrized version of Mathias forcing, we prove that there exists a countable submaximal subspace of 2ω1 ...whilst c=ω2. Combining this with previous results, we construct a disjointly tight countable irresolvable space of weight <c, answering a question of Bella and Hrušák.
A clique covering of a graph G is a set of cliques of G such that any edge of G is contained in one of these cliques, and the weight of a clique covering is the sum of the sizes of the cliques in it. ...The sigma clique cover numberscc(G) of a graph G, is defined as the smallest possible weight of a clique covering of G. Let Kt(d) denote the complete t-partite graph with each part of size d. We prove that for any fixed d≥2, we have limt→∞scc(Kt(d))=d2tlogt. This disproves a conjecture of Davoodi et al. (2016).
It is known that in X=A×B, where A and B are subspaces of ordinals, all closed C⁎-embedded subspaces of X are P-embedded. Also it is asked whether all closed C⁎-embedded subspaces of X are P-embedded ...whenever X is a subspace of products of two ordinals.
In this paper, we prove that both of the following are consistent with ZFC:•there is a subspace X of (ω+1)×ω1 such that the closed subspace X∩({ω}×ω1) is C⁎-embedded in X but not P-embedded in X,•for every subspace X of (ω+1)×ω1, if the closed subspace X∩({ω}×ω1) is C⁎-embedded in X, then it is P-embedded in X.
Let
X
be an uncountable Polish space. L̆ubica Holá showed recently that there are
2
c
quasi-continuous real valued functions defined on the uncountable Polish space
X
that are not Borel measurable. ...Inspired by Holá’s result, we are extending it in two directions. First, we prove that if
X
is an uncountable Polish space and
Y
is any Hausdorff space with
|
Y
|
≥
2
then the family of all non-Borel measurable quasi-continuous functions has cardinality
≥
2
c
. Secondly, we show that the family of quasi-continuous non Borel functions from
X
to
Y
may contain big algebraic structures.
Non-meager free sets and independent families MEDINI, ANDREA; REPOVŠ, DUŠAN; ZDOMSKYY, LYUBOMYR
Proceedings of the American Mathematical Society,
09/2017, Volume:
145, Issue:
9
Journal Article
Peer reviewed
Open access
Our main result is that, given a collection \mathcal {R} of meager relations on a Polish space X such that \vert\mathcal {R}\vert\leq \omega , there exists a dense Baire subspace F of X ...(equivalently, a nowhere meager subset F of X) such that F is R-free for every R\in \mathcal {R}. This generalizes a recent result of Banakh and Zdomskyy. As an application, we show that there exists a non-meager independent family on \omega , and define the corresponding cardinal invariant. Furthermore, assuming Martin's Axiom for countable posets, our result can be strengthened by substituting `` \vert\mathcal {R}\vert\leq \omega '' with `` \vert\mathcal {R}\vert<\mathfrak{c}'' and ``Baire'' with ``completely Baire''.
We show that for arbitrary linearly ordered set (X,≤) any bounded family of (not necessarily, continuous) real valued functions on X with bounded total variation does not contain independent ...sequences. We obtain generalized Helly's sequential compactness type theorems. One of the theorems asserts that for every compact metric space (Y,d) the compact space BVr(X,Y) of all functions X→Y with variation ≤r is sequentially compact in the pointwise topology. Another Helly type theorem shows that the compact space M+(X,Y) of all order preserving maps X→Y is sequentially compact where Y is a compact metrizable partially ordered space in the sense of Nachbin.
Properties of certain families of subsets of Euclidean spaces are established. Using the established properties theorems concerning the structure of constituents of finite independent families of ...convex bodies in R2 and R3 spaces are proved.
In this paper we will prove a slight generalisation of the Hewitt-Marczewski-Pondiczery theorem concerning the density of k-box-products. With this result we will prove the existence of generalized ...independent families of big cardinality which were introduced by Wanjun Hu.
We investigate the Boolean functions that combine various properties: the extremal values of complexity characteristics ofminimization, the inapplicability of local methods for reducing the ...complexity of the exhaustion, and the impossibility to efficiently use sufficient minimality conditions. Some quasicyclic functions are constructed that possess the properties of cyclic and zone functions, the dominance of vertex sets, and the validity of sufficient minimality conditions based on independent families of sets. For such functions, we obtain the exponential lower bounds for the extent and special sets and also a twice exponential lower bound for the number of shortest and minimal complexes of faces with distinct sets of proper vertices.
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