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A note on functional tightness and minitightness of space of the G -permutation degree

Dimitrios N. Georgiou, Nodirbek K. Mamadaliev, Rustam M. Zhuraev (2023)

Commentationes Mathematicae Universitatis Carolinae

We study the behavior of the minimal tightness and functional tightness of topological spaces under the influence of the functor of the permutation degree. Analytically: a) We introduce the notion of τ -open sets and investigate some basic properties of them. b) We prove that if the map f : X → Y is τ -continuous, then the map S P n f : S P n X → S P n Y is also τ -continuous. c) We show that the functor S P n preserves the functional tightness and the minimal tightness of compacts. d) Finally, we give some facts and properties on τ -bounded...

A note on inverse limits of continuous images of arcs.

Ivan Loncar (1999)

Publicacions Matemàtiques

The main purpose of this paper is to prove some theorems concerning inverse systems and limits of continuous images of arcs. In particular, we shall prove that if X = {Xa, pab, A} is an inverse system of continuous images of arcs with monotone bonding mappings such that cf (card (A)) ≠ w1, then X = lim X is a continuous image of an arc if and only if each proper subsystem {Xa, pab, B} of X with cf(card (B)) = w1 has the limit which is a continuous image of an arc (Theorem 18).

A note on pseudobounded paratopological groups

Fucai Lin, Shou Lin, Iván Sánchez (2014)

Topological Algebra and its Applications

Let G be a paratopological group. Then G is said to be pseudobounded (resp. ω-pseudobounded) if for every neighbourhood V of the identity e in G, there exists a natural number n such that G = Vn (resp.we have G = ∪ n∈N Vn). We show that every feebly compact (2-pseudocompact) pseudobounded (ω-pseudobounded) premeager paratopological group is a topological group. Also,we prove that if G is a totally ω-pseudobounded paratopological group such that G is a Lusin space, then is G a topological group....

A note on spaces with countable extent

Yan-Kui Song (2017)

Commentationes Mathematicae Universitatis Carolinae

Let P be a topological property. A space X is said to be star P if whenever 𝒰 is an open cover of X , there exists a subspace A ⊆ X with property P such that X = S t ( A , 𝒰 ) . In this note, we construct a Tychonoff pseudocompact SCE-space which is not star Lindelöf, which gives a negative answer to a question of Rojas-Sánchez and Tamariz-Mascarúa.

A note on topological groups and their remainders

Liang-Xue Peng, Yu-Feng He (2012)

Czechoslovak Mathematical Journal

In this note we first give a summary that on property of a remainder of a non-locally compact topological group G in a compactification b G makes the remainder and the topological group G all separable and metrizable. If a non-locally compact topological group G has a compactification b G such that the remainder b G ∖ G of G belongs to 𝒫 , then G and b G ∖ G are separable and metrizable, where 𝒫 is a class of spaces which satisfies the following conditions: (1) if X ∈ 𝒫 , then every compact subset of the space X is a...

A Ramsey theorem for polyadic spaces

Murray Bell (1996)

Fundamenta Mathematicae

A polyadic space is a Hausdorff continuous image of some power of the one-point compactification of a discrete space. We prove a Ramsey-like property for polyadic spaces which for Boolean spaces can be stated as follows: every uncountable clopen collection contains an uncountable subcollection which is either linked or disjoint. One corollary is that ( α κ ) ω is not a universal preimage for uniform Eberlein compact spaces of weight at most κ, thus answering a question of Y. Benyamini, M. Rudin and M. Wage....

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