### ${\U0001d50f}_{\U0001d520}$ und $\U0001d520$-einbettbare Limesräume

Skip to main content (access key 's'),
Skip to navigation (access key 'n'),
Accessibility information (access key '0')

Topologies τ₁ and τ₂ on a set X are called T₁-complementary if τ₁ ∩ τ₂ = X∖F: F ⊆ X is finite ∪ ∅ and τ₁∪τ₂ is a subbase for the discrete topology on X. Topological spaces $(X,{\tau}_{X})$ and $(Y,{\tau}_{Y})$ are called T₁-complementary provided that there exists a bijection f: X → Y such that ${\tau}_{X}$ and ${f}^{-1}\left(U\right):U\in {\tau}_{Y}$ are T₁-complementary topologies on X. We provide an example of a compact Hausdorff space of size ${2}^{}$ which is T₁-complementary to itself ( denotes the cardinality of the continuum). We prove that the existence of a compact Hausdorff...

We define various ring sequential convergences on $\mathbb{Z}$ and $\mathbb{Q}$. We describe their properties and properties of their convergence completions. In particular, we define a convergence ${\mathbb{L}}_{1}$ on $\mathbb{Z}$ by means of a nonprincipal ultrafilter on the positive prime numbers such that the underlying set of the completion is the ultraproduct of the prime finite fields $\mathbb{Z}/\left(p\right)$. Further, we show that $(\mathbb{Z},{\mathbb{L}}_{1}^{*})$ is sequentially precompact but fails to be strongly sequentially precompact; this solves a problem posed by D. Dikranjan.

We compare the forcing-related properties of a complete Boolean algebra $\mathbb{B}$ with the properties of the convergences ${\lambda}_{\mathrm{s}}$ (the algebraic convergence) and ${\lambda}_{\mathrm{ls}}$ on $\mathbb{B}$ generalizing the convergence on the Cantor and Aleksandrov cube, respectively. In particular, we show that ${\lambda}_{\mathrm{ls}}$ is a topological convergence iff forcing by $\mathbb{B}$ does not produce new reals and that ${\lambda}_{\mathrm{ls}}$ is weakly topological if $\mathbb{B}$ satisfies condition $\left(\hslash \right)$ (implied by the $\U0001d531$-cc). On the other hand, if ${\lambda}_{\mathrm{ls}}$ is a weakly topological convergence, then $\mathbb{B}$ is a ${2}^{\U0001d525}$-cc algebra...