Large normal ideals concentrating on a fixed small cardinality

Filters and Ideal Independence

A family $\mathscr{I} \subseteq [ω]^ω$ such that for all finite $\{X_i\}_{i\in n}\subseteq \mathcal I$ and $A \in \mathscr{I} \setminus \{X_i\}_{i\in n}$, the set $A \setminus \bigcup_{i < n} X_i$ is infinite, is said to be ideal independent. An ideal independent family which is maximal under inclusion is said to be a maximal ideal independent family and the least cardinality of such family is denoted $\mathfrak{s}_{mm}$. We show that $\mathfrak{u}\leq\mathfrak{s}_{mm}$, which in particular establishes the independence of $\mathfrak{s}_{mm}$ and $\mathfrak{i}$. Given an arbitrary set $C$ of un…

On the minimal cardinality of a subset of R which is not of first category

Let M be the ideal of first category subsets of R and non(M)=min{card X: X \subseteq R, X \not\in M}. We consider families Φof sequences converging to \infty, with the property that for every open set U \subseteq R that is unbounded above there exists a sequence belonging to Φ, which has an infinite number of terms belonging to U. We present assumptions about Φwhich imply that the minimal cardinality of Φequals non(M).

A new combinatorial characterization of the minimal cardinality of a subset of R which is not of first category

Let M denote the ideal of first category subsets of R. We prove that min{card X: X \subseteq R, X \not\in M} is the smallest cardinality of a family S \subseteq {0,1}^ωwith the property that for each f: ω-> \bigcup_{n \in ω}{0,1}^n there exists a sequence {a_n}_{n \in ω} belonging to S such that for infinitely many i \in ωthe infinite sequence {a_{i+n}}_{n \in ω} extends the finite sequence f(i). We inform that S \subseteq {0,1}^ωis not of first category if and only if for each f: ω-> \bigcup_{n \in ω}{0,1}^n there exists a sequence {a_n}_{n \in ω} belonging to S such that for infinitely many …

滤子、理想独立性与理想Mrówka空间

Filters, ideal independence and ideal Mrówka spaces

本文研究了理想独立族，证明了极大理想独立族的等价条件，确立了基数不变量间的独立性与一致性，并引入了相关的自然滤子与Mrówka空间类比。

On large cardinals and generalized Baire spaces

Working under large cardinal assumptions, we study the Borel-reducibility between equivalence relations modulo restrictions of the non-stationary ideal on some fixed cardinal $κ$. We show the consistency of $E^{λ^{++},λ^{++}}_{λ\text{-club}}$, the relation of equivalence modulo the non-stationary ideal restricted to $S^{λ^{++}}_λ$ in the space $(λ^{++})^{λ^{++}}$, being continuously reducible to $E^{2,λ^{++}}_{λ^+\text{-club}}$, the relation of equivalence modulo the non-stationary ideal restricted to $S^{λ^{++}}_{λ^+}$ in the space $2^{λ^{++}}$. Then we show the consistency of $E^{2,κ}_{reg}$…

通过无限基数的幂等性研究群与环的基数

Cardinality of groups and rings via the idempotency of infinite cardinals

本文利用 ZFC 中无限基数幂等性这一集合论事实，研究群与环的基数性质，证明了无限阿贝尔群具有同基数真子群的充要条件，计算了幺半群环与多项式环的基数，并引入平衡环概念，刻画了诺特环与自内射环成为平衡环的条件。