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On linear operators and functors extending pseudometrics

C. Bessaga (1993)

Fundamenta Mathematicae

For some pairs (X,A), where X is a metrizable topological space and A its closed subset, continuous, linear (i.e., additive and positive-homogeneous) operators extending metrics for A to metrics for X are constructed. They are defined by explicit analytic formulas, and also regarded as functors between certain categories. An essential role is played by "squeezed cones" related to the classical cone construction. The main result: if A is a nondegenerate absolute neighborhood retract for metric spaces,...

On meager function spaces, network character and meager convergence in topological spaces

Taras O. Banakh, Volodymyr Mykhaylyuk, Lubomyr Zdomsky (2011)

Commentationes Mathematicae Universitatis Carolinae

For a non-isolated point x of a topological space X let nw χ ( x ) be the smallest cardinality of a family 𝒩 of infinite subsets of X such that each neighborhood O ( x ) ⊂ X of x contains a set N ∈ 𝒩 . We prove that (a) each infinite compact Hausdorff space X contains a non-isolated point x with nw χ ( x ) = ℵ 0 ; (b) for each point x ∈ X with nw χ ( x ) = ℵ 0 there is an injective sequence ( x n ) n ∈ ω in X that ℱ -converges to x for some meager filter ℱ on ω ; (c) if a functionally Hausdorff space X contains an ℱ -convergent injective sequence for some meager filter...

On n -in-countable bases

S. A. Peregudov (2000)

Commentationes Mathematicae Universitatis Carolinae

Some results concerning spaces with countably weakly uniform bases are generalized for spaces with n -in-countable ones.

On ( n , m ) - A -normal and ( n , m ) - A -quasinormal semi-Hilbertian space operators

Samir Al Mohammady, Sid Ahmed Ould Beinane, Sid Ahmed Ould Ahmed Mahmoud (2022)

Mathematica Bohemica

The purpose of the paper is to introduce and study a new class of operators on semi-Hilbertian spaces, i.e. spaces generated by positive semi-definite sesquilinear forms. Let ℋ be a Hilbert space and let A be a positive bounded operator on ℋ . The semi-inner product 〈 h ∣ k 〉 A : = 〈 A h ∣ k 〉 , h , k ∈ ℋ , induces a semi-norm ∥ · ∥ A . This makes ℋ into a semi-Hilbertian space. An operator T ∈ ℬ A ( ℋ ) is said to be ( n , m ) - A -normal if [ T n , ( T ♯ A ) m ] : = T n ( T ♯ A ) m - ( T ♯ A ) m T n = 0 for some positive integers n and m .

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