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𝒵 -distributive function lattices

Marcel Erné (2013)

Mathematica Bohemica

It is known that for a nonempty topological space X and a nonsingleton complete lattice Y endowed with the Scott topology, the partially ordered set [ X , Y ] of all continuous functions from X into Y is a continuous lattice if and only if both Y and the open set lattice 𝒪 X are continuous lattices. This result extends to certain classes of 𝒵 -distributive lattices, where 𝒵 is a subset system replacing the system 𝒟 of all directed subsets (for which the 𝒟 -distributive complete lattices are just the continuous...

Bohr compactifications of discrete structures

Joan Hart, Kenneth Kunen (1999)

Fundamenta Mathematicae

We prove the following theorem: Given a⊆ω and 1 ≤ α < ω 1 C K , if for some η < ℵ 1 and all u ∈ WO of length η, a is Σ α 0 ( u ) , then a is Σ α 0 .We use this result to give a new, forcing-free, proof of Leo Harrington’s theorem: Σ 1 1 -Turing-determinacy implies the existence of 0 .

Connected LCA groups are sequentially connected

Shou Lin, Mihail G. Tkachenko (2013)

Commentationes Mathematicae Universitatis Carolinae

We prove that every connected locally compact Abelian topological group is sequentially connected, i.e., it cannot be the union of two proper disjoint sequentially closed subsets. This fact is then applied to the study of extensions of topological groups. We show, in particular, that if H is a connected locally compact Abelian subgroup of a Hausdorff topological group G and the quotient space G / H is sequentially connected, then so is G .

Hopfian and co-Hopfian objects.

Kalathoor Varadarajan (1992)

Publicacions Matemàtiques

The aim of the present paper is to study Hopfian and Co-Hopfian objects in categories like the category of rings, the module categories A-mod and mod-A for any ring A. Using Stone's representation theorem any Boolean ring can be regarded as the ring A of clopen subsets of compact Hausdorff totally disconnected space X. It turns out that the Boolean ring A will be Hopfian (resp. co-Hopfian) if and only if the space X is co-Hopfian (resp. Hopfian) in the category Top. For any compact Hausdorff space...

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