Compact Group Topologies for R

Topological Group Criterion for C(x) in Compact-open-like Topologies, I

We address questions of when (C(X), +) is a topological group in some topologies which are meets of systems of compact-open topologies from certain dense subsets of X. These topologies have arisen from the theory of epimorphisms in lattice-ordered groups (in this context called “epi-topology”). A basic necessary and sufficient condition is developed, which at least yields enough insight to prov...

Homomorphisms on Topological Groups from the Perspective of Bourbaki-boundedness

In this note we study some topological properties of bounded sets and Bourbaki-bounded sets. Also we introduce two types of Bourbaki-bounded homomorphisms on topological groups  including, n$-$Bourbaki-bounded homomorphisms and$hspace{1mm}$ B$-$Bourbaki-bounded homomorphisms. We compare them to each other and with the class of continuous homomorphisms. So, two topologies are presented on them a...

Reiter’s Properties for the Actions of Locally Compact Quantum Goups on von Neumann Algebras

Maximal and Minimal Topologieso

A topological space (X, T) with property R is maximal R (minimal R) if T is a maximal (minimal) element in the set R(X) of all topologies on the set X having property R with the partial ordering of set inclusions. The properties of maximal topologies for compactness, countable compactness, sequential compactness, Bolzano-Weierstrass compactness, and Lindelöf are investigated and the relations b...

Shift Invariant Spaces and Shift Preserving Operators on Locally Compact Abelian Groups

We investigate shift invariant subspaces of $L^2(G)$, where $G$ is a locally compact abelian group. We show that every shift invariant space can be decomposed as an orthogonal sum of spaces each of which is generated by a single function whose shifts form a Parseval frame. For a second countable locally compact abelian group $G$ we prove a useful Hilbert space isomorphism, introduce range funct...