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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 $(X, τ_{X})$ and $(Y, τ_{Y})$ are called T₁-complementary provided that there exists a bijection f: X → Y such that $τ_{X}$ and $f^{- 1} (U) : U \in τ_{Y}$ 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...

A completion functor for Cauchy groups.

Fric, R., Kent, Darrell C. (1981)

International Journal of Mathematics and Mathematical Sciences

A completion of $ℤ$ is a field

José E. Marcos (2003)

Czechoslovak Mathematical Journal

We define various ring sequential convergences on $ℤ$ and $ℚ$ . We describe their properties and properties of their convergence completions. In particular, we define a convergence $𝕃_{1}$ on $ℤ$ 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 $ℤ / (p)$ . Further, we show that $(ℤ, 𝕃_{1}^{*})$ is sequentially precompact but fails to be strongly sequentially precompact; this solves a problem posed by D. Dikranjan.

A convergence on Boolean algebras generalizing the convergence on the Aleksandrov cube

Miloš S. Kurilić, Aleksandar Pavlović (2014)

Czechoslovak Mathematical Journal

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