Embedding compact t-semigroups into compact uniquely divisible semigroups

Communicated by J. D. Lawson     

A compact group which is not Valdivia compact

A compact space is Valdivia compact if it can be embedded in a Tikhonov cube in such a way that the intersection is dense in , where is the sigma-product ( the set of points with countably many non-zero coordinates). We show that there exists a compact connected Abelian group of weight which is not Valdivia compact, and deduce that Valdivia compact spaces are not preserved by open maps.

     

A note on locally compact subsemigroups of compact groups

Semigroup Forum - An elementary proof is given for the fact that every locally compact subsemigroup of a compact topological group is a closed subgroup. A sample consequence is that every...     

A normal countably compact not absolutely countably compact space

We construct an example of a normal countably compact not absolutely countably compact space. We also prove that every hereditarily normal countably compact space is absolutely countably compact and suggest a method for construction of hereditarily normal spaces without property .

     

Hilbert compacts,compact ellipsoids,and compact extrema

We consider a system of so-called Hilbert compacts K(H) in a Hilbert space H; those Hilbert compacts admit a two-sided estimate by compact ellipsoids in H. For functionals in H, we introduce the notion of a compact extremum achieved at a certain base with respect to the imbedding in K(H). An example of the K-extremum of a variational functional in the Sobolev space W ₂¹ is considered.     

Torsion-complete and algebraically compact groups with compact endomorphism rings

We give a complete characterization of torsion-complete and algebraically compact abelian groups whose endomorphism rings admit a compact ring topology.     

Both compact and sequentially compact sets in abelian topological group

We show that every abelian topological group contains many interesting sets which are both compact and sequentially compact. Then we can deduce some useful facts, e.g.,

(1)

if G is a Hausdorff abelian topological group and μ:N²→G is countably additive, then the range μ(N²)={μ(A):A⊆N} is compact metrizable;

(2)

if X is a Hausdorff locally convex space and {xj}⊂X, then F={_∑j∈Δxj:Δ⊂N, Δ is finite} is relatively compact in (X,weak) if and only if F is relatively compact in X, and if and only if F is relatively compact in (X,F(M)) where F(M) is the Dierolf topology which is the strongest 〈X,X^′〉-polar topology having the same subseries convergent series as the weak topology.

     

Lipschitz compact operators

We introduce the notion of Lipschitz compact (weakly compact, finite-rank, approximable) operators from a pointed metric space X into a Banach space E. We prove that every strongly Lipschitz p-nuclear operator is Lipschitz compact and every strongly Lipschitz p-integral operator is Lipschitz weakly compact. A theory of Lipschitz compact (weakly compact, finite-rank) operators which closely parallels the theory for linear operators is developed. In terms of the Lipschitz transpose map of a Lipschitz operator, we state Lipschitz versions of Schauder type theorems on the (weak) compactness of the adjoint of a (weakly) compact linear operator.     

Semimodular compact groups

A study of compact groups in which the lattice of all closed subgroups satisfies one of the following requirements: the Jordan-Dedekind condition for chains; semimodularity below; the same plus the existence of complements; the same plus the existence of relative complements —is presented.Translated from Matematicheskie Zametki, Vol. 3, No. 5, pp. 503–510, May, 1968.     

Compact spreads and compact translation planes over locally compact fields

We prove that a spread S over a locally compact nondlscrete field F defines a topological translation plane if and only if the spread is compact. For F=R, this is implicit in Breuning's thesis [Bre], cf. [B 2]. For the proof, we describe the point set of the projective translation plane as a quotient space of some projective space, with identifications taking place in one hyperplane. This is new even for F=R.