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A nice subclass of functionally countable spaces

Vladimir Vladimirovich Tkachuk — 2018

Commentationes Mathematicae Universitatis Carolinae

A space X is functionally countable if f ( X ) is countable for any continuous function f : X . We will call a space X exponentially separable if for any countable family of closed subsets of X , there exists a countable set A X such that A 𝒢 whenever 𝒢 and 𝒢 . Every exponentially separable space is functionally countable; we will show that for some nice classes of spaces exponential separability coincides with functional countability. We will also establish that the class of exponentially separable spaces has...

A note on splittable spaces

Vladimir Vladimirovich Tkachuk — 1992

Commentationes Mathematicae Universitatis Carolinae

A space X is splittable over a space Y (or splits over Y ) if for every A X there exists a continuous map f : X Y with f - 1 f A = A . We prove that any n -dimensional polyhedron splits over 𝐑 2 n but not necessarily over 𝐑 2 n - 2 . It is established that if a metrizable compact X splits over 𝐑 n , then dim X n . An example of n -dimensional compact space which does not split over 𝐑 2 n is given.

Some non-multiplicative properties are l -invariant

Vladimir Vladimirovich Tkachuk — 1997

Commentationes Mathematicae Universitatis Carolinae

A cardinal function ϕ (or a property 𝒫 ) is called l -invariant if for any Tychonoff spaces X and Y with C p ( X ) and C p ( Y ) linearly homeomorphic we have ϕ ( X ) = ϕ ( Y ) (or the space X has 𝒫 ( X 𝒫 ) iff Y 𝒫 ). We prove that the hereditary Lindelöf number is l -invariant as well as that there are models of Z F C in which hereditary separability is l -invariant.

Some new versions of an old game

Vladimir Vladimirovich Tkachuk — 1995

Commentationes Mathematicae Universitatis Carolinae

The old game is the point-open one discovered independently by F. Galvin [7] and R. Telgársky [17]. Recall that it is played on a topological space X as follows: at the n -th move the first player picks a point x n X and the second responds with choosing an open U n x n . The game stops after ω moves and the first player wins if { U n : n ω } = X . Otherwise the victory is ascribed to the second player. In this paper we introduce and study the games θ and Ω . In θ the moves are made exactly as in the point-open game, but the...

A nice class extracted from C p -theory

Vladimir Vladimirovich Tkachuk — 2005

Commentationes Mathematicae Universitatis Carolinae

We study systematically a class of spaces introduced by Sokolov and call them Sokolov spaces. Their importance can be seen from the fact that every Corson compact space is a Sokolov space. We show that every Sokolov space is collectionwise normal, ω -stable and ω -monolithic. It is also established that any Sokolov compact space X is Fréchet-Urysohn and the space C p ( X ) is Lindelöf. We prove that any Sokolov space with a G δ -diagonal has a countable network and obtain some cardinality restrictions on subsets...

Diagonals and discrete subsets of squares

Dennis BurkeVladimir Vladimirovich Tkachuk — 2013

Commentationes Mathematicae Universitatis Carolinae

In 2008 Juhász and Szentmiklóssy established that for every compact space X there exists a discrete D X × X with | D | = d ( X ) . We generalize this result in two directions: the first one is to prove that the same holds for any Lindelöf Σ -space X and hence X ω is d -separable. We give an example of a countably compact space X such that X ω is not d -separable. On the other hand, we show that for any Lindelöf p -space X there exists a discrete subset D X × X such that Δ = { ( x , x ) : x X } D ¯ ; in particular, the diagonal Δ is a retract of D ¯ and the projection...

Some applications of the point-open subbase game

D. Guerrero SánchezVladimir Vladimirovich Tkachuk — 2017

Commentationes Mathematicae Universitatis Carolinae

Given a subbase 𝒮 of a space X , the game P O ( 𝒮 , X ) is defined for two players P and O who respectively pick, at the n -th move, a point x n X and a set U n 𝒮 such that x n U n . The game stops after the moves { x n , U n : n ø } have been made and the player P wins if n ø U n = X ; otherwise O is the winner. Since P O ( 𝒮 , X ) is an evident modification of the well-known point-open game P O ( X ) , the primary line of research is to describe the relationship between P O ( X ) and P O ( 𝒮 , X ) for a given subbase 𝒮 . It turns out that, for any subbase 𝒮 , the player P has a winning strategy...

Addition theorems, D -spaces and dually discrete spaces

Ofelia Teresa AlasVladimir Vladimirovich TkachukRichard Gordon Wilson — 2009

Commentationes Mathematicae Universitatis Carolinae

A in a space X is a family 𝒪 = { O x : x X } of open subsets of X such that x O x for any x X . A set Y X is if 𝒪 ( Y ) = { O x : x Y } = X . If every neighbourhood assignment in X has a closed and discrete (respectively, discrete) kernel, then X is said to be a D -space (respectively a dually discrete space). In this paper we show among other things that every GO-space is dually discrete, every subparacompact scattered space and every continuous image of a Lindelöf P -space is a D -space and we prove an addition theorem for metalindelöf spaces which...

Čech-completeness and ultracompleteness in “nice spaces”

Miguel López de LunaVladimir Vladimirovich Tkachuk — 2002

Commentationes Mathematicae Universitatis Carolinae

We prove that if X n is a union of n subspaces of pointwise countable type then the space X is of pointwise countable type. If X ω is a countable union of ultracomplete spaces, the space X ω is ultracomplete. We give, under CH, an example of a Čech-complete, countably compact and non-ultracomplete space, giving thus a partial answer to a question asked in [BY2].

A quest for nice kernels of neighbourhood assignments

Raushan Z. BuzyakovaVladimir Vladimirovich TkachukRichard Gordon Wilson — 2007

Commentationes Mathematicae Universitatis Carolinae

Given a topological property (or a class) 𝒫 , the class 𝒫 * dual to 𝒫 (with respect to neighbourhood assignments) consists of spaces X such that for any neighbourhood assignment { O x : x X } there is Y X with Y 𝒫 and { O x : x Y } = X . The spaces from 𝒫 * are called . We continue the study of this duality which constitutes a development of an idea of E. van Douwen used to define D -spaces. We prove a number of results on duals of some general classes of spaces establishing, in particular, that any generalized ordered space of countable...

In quest of weaker connected topologies

Mihail G. TkachenkoVladimir Vladimirovich TkachukVladimir Vladimirovich UspenskijRichard Gordon Wilson — 1996

Commentationes Mathematicae Universitatis Carolinae

We study when a topological space has a weaker connected topology. Various sufficient and necessary conditions are given for a space to have a weaker Hausdorff or regular connected topology. It is proved that the property of a space of having a weaker Tychonoff topology is preserved by any of the free topological group functors. Examples are given for non-preservation of this property by “nice” continuous mappings. The requirement that a space have a weaker Tychonoff connected topology is rather...

Connectedness and local connectedness of topological groups and extensions

Ofelia Teresa AlasMihail G. TkachenkoVladimir Vladimirovich TkachukRichard Gordon Wilson — 1999

Commentationes Mathematicae Universitatis Carolinae

It is shown that both the free topological group F ( X ) and the free Abelian topological group A ( X ) on a connected locally connected space X are locally connected. For the Graev’s modification of the groups F ( X ) and A ( X ) , the corresponding result is more symmetric: the groups F Γ ( X ) and A Γ ( X ) are connected and locally connected if X is. However, the free (Abelian) totally bounded group F T B ( X ) (resp., A T B ( X ) ) is not locally connected no matter how “good” a space X is. The above results imply that every non-trivial continuous homomorphism...

On dense subspaces satisfying stronger separation axioms

We prove that it is independent of ZFC whether every Hausdorff countable space of weight less than c has a dense regular subspace. Examples are given of countable Hausdorff spaces of weight c which do not have dense Urysohn subspaces. We also construct an example of a countable Urysohn space, which has no dense completely Hausdorff subspace. On the other hand, we establish that every Hausdorff space of π -weight less than 𝔭 has a dense completely Hausdorff (and hence Urysohn) subspace. We show that...

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