We extend the idea of $I$-convergence and $I^{*}$-convergence of sequences to a topological space and derive several basic properties of these concepts in the topological space.

Based on some earlier findings on Banach Category Theorem for some “nice” $\sigma$-ideals by J. Kaniewski, D. Rose and myself I introduce the $h$ operator ($h$ stands for “heavy points”) to refine and generalize kernel constructions of A. H. Stone. Having obtained in this way a generalized Kuratowski’s decomposition theorem I prove some characterizations of the domains of functions having “many” points of $h$-continuity. Results of this type lead, in the case of the $\sigma$-ideal of meager sets, to important statements...

In this paper we introduce the $ℐ$- and ${ℐ}^{*}$-convergence and divergence of nets in $\left(\ell \right)$-groups. We prove some theorems relating different types of convergence/divergence for nets in $\left(\ell \right)$-group setting, in relation with ideals. We consider both order and $\left(D\right)$-convergence. By using basic properties of order sequences, some fundamental properties, Cauchy-type characterizations and comparison results are derived. We prove that ${ℐ}^{*}$-convergence/divergence implies $ℐ$-convergence/divergence for every ideal, admissible for...

We make use of a forcing technique for extending Boolean algebras. The same type of forcing was employed in Baumgartner J.E., Komjáth P., Boolean algebras in which every chain and antichain is countable, Fund. Math. 111 (1981), 125–133, Koszmider P., Forcing minimal extensions of Boolean algebras, Trans. Amer. Math. Soc. 351 (1999), no. 8, 3073–3117, and elsewhere. Using and modifying a lemma of Koszmider, and using CH, we obtain an atomless BA, $A$ such that $𝔣\left(A\right)={\text{s}}_{\text{mm}}\left(A\right)<𝔲\left(A\right)$, answering questions raised by Monk...

We consider various forms of Ramsey's theorem, the monotone subsequence theorem and the Bolzano-Weierstrass theorem which are connected with ideals of subsets of natural numbers. We characterize ideals with properties considered. We show that, in a sense, Ramsey's theorem, the monotone subsequence theorem and the Bolzano-Weierstrass theorem characterize the same class of ideals. We use our results to show some versions of density Ramsey's theorem (these are similar to generalizations shown in [P....

Given a topological space ⟨X,T⟩ ∈ M, an elementary submodel of set theory, we define $X_M$ to be X ∩ M with topology generated by U ∩ M:U ∈ T ∩ M. We prove that if $X_M$ is homeomorphic to ℝ, then $X=X_M$. The same holds for arbitrary locally compact uncountable separable metric spaces, but is independent of ZFC if “local compactness” is omitted.

The least cardinal λ such that some (equivalently: every) compact group with weight α admits a dense, pseudocompact subgroup of cardinality λ is denoted by m(α). Clearly, $m\left(\alpha \right)\le 2^{\alpha }$. We show:    Theorem 4.12. Let G be Abelian with |G| = γ. If either m(α) ≤ α and m$\left(\alpha \right)\le r_0\left(G\right)\le \gamma \le 2^{\alpha }$, or α > ω and ${\alpha }^{\omega }\le r_0\left(G\right)\le 2^{\alpha }$, then G admits a pseudocompact group topology of weight α.  Theorem 4.15. Every connected, pseudocompact Abelian group G with wG = α ≥ ω satisfies $r_0\left(G\right)\ge m\left(\alpha \right)$.  Theorem 5.2(b). If G is divisible Abelian with $2^{r_0\left(G\right)}\le \gamma$, then G admits at most $2^{\gamma }$-many...

Let $L^{\varphi }$ be an Orlicz space defined by a convex Orlicz function $\varphi$ and let $E^{\varphi }$ be the space of finite elements in $L^{\varphi }$ (= the ideal of all elements of order continuous norm). We show that the usual norm topology ${𝒯}_{\varphi }$ on $L^{\varphi }$ restricted to $E^{\varphi }$ can be obtained as an inductive limit topology with respect to some family of other Orlicz spaces. As an application we obtain a characterization of continuity of linear operators defined on $E^{\varphi }$.

We apply the theory of infinite two-person games to two well-known problems in topology: Suslin’s Problem and Arhangel’skii’s problem on the weak Lindelöf number of the $G_{\delta }$ topology on a compact space. More specifically, we prove results of which the following two are special cases: 1) every linearly ordered topological space satisfying the game-theoretic version of the countable chain condition is separable, and 2) in every compact space satisfying the game-theoretic version of the weak Lindelöf...

This work presents some cardinal inequalities in which appears the closed pseudo-character, ${\psi }_c$, of a space. Using one of them — ${\psi }_c\left(X\right)\le 2^{d\left(X\right)}$ for $T_2$ spaces — we improve, from $T_3$ to $T_2$ spaces, the well-known result that initially $\kappa$-compact $T_3$ spaces are $\lambda$-bounded for all cardinals $\lambda$ such that $2^{\lambda }\le \kappa$. And then, using an idea of A. Dow, we prove that initially $\kappa$-compact $T_2$ spaces are in fact compact for $\kappa =2^{F\left(X\right)}$, $2^{s\left(X\right)}$, $2^{t\left(X\right)}$, $2^{\chi \left(X\right)}$, $2^{{\psi }_c\left(X\right)}$ or $\kappa =max\{{\tau }^{+},{\tau }^{<\tau }\}$, where $\tau >t(p,X)$ for all $p\in X$.