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Recognizing the topologies of spaces of metrics with the topology of uniform convergence

Koshino Katsuhisa

Bulletin of the Polish Academy of Sciences. Mathematics

2022

Vol. 70, no. 2

165--171

Given a metrizable space X of density κ, we study the topological structure of the space PM(X) of continuous bounded pseudometrics on X, which is endowed with the topology of uniform convergence. We prove that PM(X) is homeomorphic to [0,1)κ(κ−1)/2 if X is finite, to ℓ2(2<κ) if X is infinite and generalized compact, and to ℓ2(2κ) if X is not generalized compact. We also show that for an infinite σ-compact metrizable space X, the space M(X)⊂PM(X) of continuous bounded metrics on X and the space AM(X)⊂M(X) of bounded admissible metrics on X are homeomorphic to ℓ2 if X is compact, and to ℓ∞ if X is not compact.

On ∞-entropy points in real analysis

Korczak-Kubiak E., Loranty A., Pawlak R. J.

Opuscula Mathematica

2014

Vol. 34, no. 4

799--812

We will consider ∞-entropy points in the context of the possibilities of approximation mappings by the functions having ∞-entropy points and belonging to essential (from the point of view of real analysis theory) classes of functions: almost continuous, Darboux Baire one and approximately continuous functions.

Extreme points and support points of a class of analytic functions with missing coefficients

Xiong L., Tian L, Luo S., Liu X.

Demonstratio Mathematica

2013

Vol. 46, nr 3

525--532

Let Mn(a, b, c) denote a class of functions of the form (...) which are analytic in open unit disk (...) and satisfy the condition (...). In this paper, we obtain the extreme points and support points of the class Mn(a, b, c) of functions.

Closedness of certain classes of functions in the topology of uniform convergence

Kohli J. K., Aggarwal J.

Demonstratio Mathematica

2012

Vol. 45, nr 4

947-952

In this paper, closedness of certain classes of functions in VX in the topology of uniform convergence is observed. In particular, we show that the function spaces SC(X, Y) of quasi continuous (…) functions, (…) (X, Y ) of (…)-continuous functions and L(X,Y) of cl-supercontinuous functions are closed in YX in the topology of uniform convergence.