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13 pages, 316 KiB

Open AccessArticle

On the Čech-Completeness of the Space of τ-Smooth Idempotent Probability Measures

by Ljubiša D. R. Kočinac, Adilbek A. Zaitov and Muzaffar R. Eshimbetov

Axioms 2024, 13(8), 569; https://doi.org/10.3390/axioms13080569 - 21 Aug 2024

Viewed by 860

For the set

I (X)

of probability measures on a compact Hausdorff space X, we propose a new way to introduce the topology by using the open subsets of the space X. Then, among other things, we give a [...] Read more.

For the set

I (X)

of probability measures on a compact Hausdorff space X, we propose a new way to introduce the topology by using the open subsets of the space X. Then, among other things, we give a new proof that for a compact Hausdorff space X, the space

I (X)

is also a compact Hausdorff space. For a Tychonoff space X, we consider the topological space

I_{τ} (X)

τ

-smooth idempotent probability measures on X and show that the space

I_{τ} (X)

is Čech-complete if and only if the given space X is Čech-complete. Full article

(This article belongs to the Special Issue Topics in General Topology and Applications)

10 pages, 284 KiB

Open AccessArticle

Baire 1 Functions and the Topology of Uniform Convergence on Compacta

by Ľubica Holá and Dušan Holý

Mathematics 2024, 12(10), 1494; https://doi.org/10.3390/math12101494 - 10 May 2024

Viewed by 983

Abstract

Let X be a Tychonoff topological space,

B_{1} (X, R)

be the space of real-valued Baire 1 functions on X and

τ_{U C}

be the topology of uniform convergence on compacta. The main purpose of this paper is [...] Read more.

Let X be a Tychonoff topological space,

B_{1} (X, R)

be the space of real-valued Baire 1 functions on X and

τ_{U C}

be the topology of uniform convergence on compacta. The main purpose of this paper is to study cardinal invariants of

(B_{1} (X, R), τ_{U C})

. We prove that the following conditions are equivalent: (1)

(B_{1} (X, R), τ_{U C})

is metrizable; (2)

(B_{1} (X, R), τ_{U C})

is completely metrizable; (3)

(B_{1} (X, R), τ_{U C})

is Čech-complete; and (4) X is hemicompact. It is also proven that if X is a separable metric space with a non isolated point, then the topology of uniform convergence on compacta on

B_{1} (X, R)

is seen to behave like a metric topology in the sense that the weight, netweight, density, Lindelof number and cellularity are all equal for this topology and they are equal to

c = | B_{1} (X, R) |

. We find further conditions on X under which these cardinal invariants coincide on

B_{1} (X, R)

. Full article

(This article belongs to the Section B: Geometry and Topology)

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