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Sur la caractérisation topologique des compacts à l'aide des demi-treillis des pseudométriques continues

Taras Banakh — 1995

Studia Mathematica

For a Tikhonov space X we denote by Pc(X) the semilattice of all continuous pseudometrics on X. It is proved that compact Hausdorff spaces X and Y are homeomorphic if and only if there is a positive-homogeneous (or an additive) semi-lattice isomorphism T:Pc(X) → Pc(Y). A topology on Pc(X) is called admissible if it is intermediate between the compact-open and pointwise topologies on Pc(X). Another result states that Tikhonov spaces X and Y are homeomorphic if and only if there exists a positive-homogeneous...

Topological classification of strong duals to nuclear (LF)-spaces

Taras Banakh — 2000

Studia Mathematica

We show that the strong dual X’ to an infinite-dimensional nuclear (LF)-space is homeomorphic to one of the spaces: ω , , Q × , ω × , or ( ) ω , where = l i m n and Q = [ - 1 , 1 ] ω . In particular, the Schwartz space D’ of distributions is homeomorphic to ( ) ω . As a by-product of the proof we deduce that each infinite-dimensional locally convex space which is a direct limit of metrizable compacta is homeomorphic either to or to Q × . In particular, the strong dual to any metrizable infinite-dimensional Montel space is homeomorphic either...

Algebra in the superextensions of twinic groups

Given a group X we study the algebraic structure of the compact right-topological semigroup λ(X) consisting of all maximal linked systems on X. This semigroup contains the semigroup β(X) of ultrafilters as a closed subsemigroup. We construct a faithful representation of the semigroup λ(X) in the semigroup ( X ) ( X ) of all self-maps of the power-set (X) and show that the image of λ(X) in ( X ) ( X ) coincides with the semigroup E n d λ ( ( X ) ) of all functions f: (X) → (X) that are equivariant, monotone and symmetric in the sense...

Interplay between strongly universal spaces and pairs

Given a pair (M,X) of spaces we investigate the connections between the (strong) universality of (M,X) and that of the space X. We apply this to prove Enlarging, Deleting, and Strong Negligibility Theorems for strongly universal and absorbing spaces. Given an absorbing space Ω we also study the question of topological uniqueness of the pair (M,X), where M = [ 0 , 1 ] ω or M = ( 0 , 1 ) ω and X is a copy of Ω in M having a locally homotopy negligible complement in M.

General position properties in fiberwise geometric topology

General position properties play a crucial role in geometric and infinite-dimensional topologies. Often such properties provide convenient tools for establishing various universality results. One of well-known general position properties is DDⁿ, the property of disjoint n-cells. Each Polish L C n - 1 -space X possessing DDⁿ contains a topological copy of each n-dimensional compact metric space. This fact implies, in particular, the classical Lefschetz-Menger-Nöbeling-Pontryagin-Tolstova embedding theorem...

Perfectly supportable semigroups are σ-discrete in each Hausdorff shift-invariant topology

Taras BanakhIgor Guran — 2013

Topological Algebra and its Applications

In this paper we introduce perfectly supportable semigroups and prove that they are σ-discrete in each Hausdorff shiftinvariant topology. The class of perfectly supportable semigroups includes each semigroup S such that FSym(X) ⊂ S ⊂ FRel(X) where FRel(X) is the semigroup of finitely supported relations on an infinite set X and FSym(X) is the group of finitely supported permutations of X.

Topological groups and convex sets homeomorphic to non-separable Hilbert spaces

Taras BanakhIgor Zarichnyy — 2008

Open Mathematics

Let X be a topological group or a convex set in a linear metric space. We prove that X is homeomorphic to (a manifold modeled on) an infinite-dimensional Hilbert space if and only if X is a completely metrizable absolute (neighborhood) retract with ω-LFAP, the countable locally finite approximation property. The latter means that for any open cover 𝒰 of X there is a sequence of maps (f n: X → X)nεgw such that each f n is 𝒰 -near to the identity map of X and the family f n(X)n∈ω is locally finite...

Spaces with fibered approximation property in dimension n

Taras BanakhVesko Valov — 2010

Open Mathematics

A metric space M is said to have the fibered approximation property in dimension n (briefly, M ∈ FAP(n)) if for any ɛ > 0, m ≥ 0 and any map g: 𝕀 m × 𝕀 n → M there exists a map g′: 𝕀 m × 𝕀 n → M such that g′ is ɛ-homotopic to g and dim g′ (z × 𝕀 n) ≤ n for all z ∈ 𝕀 m. The class of spaces having the FAP(n)-property is investigated in this paper. The main theorems are applied to obtain generalizations of some results due to Uspenskij [11] and Tuncali-Valov [10].

Constructing universally small subsets of a given packing index in Polish groups

Taras BanakhNadya Lyaskovska — 2011

Colloquium Mathematicae

A subset of a Polish space X is called universally small if it belongs to each ccc σ-ideal with Borel base on X. Under CH in each uncountable Abelian Polish group G we construct a universally small subset A₀ ⊂ G such that |A₀ ∩ gA₀| = for each g ∈ G. For each cardinal number κ ∈ [5,⁺] the set A₀ contains a universally small subset A of G with sharp packing index p a c k ( A κ ) = s u p | | : g A g G i s d i s j o i n t equal to κ.

Classifying homogeneous ultrametric spaces up to coarse equivalence

Taras BanakhDušan Repovš — 2016

Colloquium Mathematicae

For every metric space X we introduce two cardinal characteristics c o v ( X ) and c o v ( X ) describing the capacity of balls in X. We prove that these cardinal characteristics are invariant under coarse equivalence, and that two ultrametric spaces X,Y are coarsely equivalent if c o v ( X ) = c o v ( X ) = c o v ( Y ) = c o v ( Y ) . This implies that an ultrametric space X is coarsely equivalent to an isometrically homogeneous ultrametric space if and only if c o v ( X ) = c o v ( X ) . Moreover, two isometrically homogeneous ultrametric spaces X,Y are coarsely equivalent if and only if c o v ( X ) = c o v ( Y ) ...

On universality of finite powers of locally path-connected meager spaces

Taras BanakhRobert Cauty — 2005

Colloquium Mathematicae

It is shown that for every integer n the (2n+1)th power of any locally path-connected metrizable space of the first Baire category is 𝓐₁[n]-universal, i.e., contains a closed topological copy of each at most n-dimensional metrizable σ-compact space. Also a one-dimensional σ-compact absolute retract X is found such that the power X^{n+1} is 𝓐₁[n]-universal for every n.

Topological classification of closed convex sets in Fréchet spaces

Taras BanakhRobert Cauty — 2011

Studia Mathematica

We prove that each non-separable completely metrizable convex subset of a Fréchet space is homeomorphic to a Hilbert space. This resolves a more than 30 years old problem of infinite-dimensional topology. Combined with the topological classification of separable convex sets due to Klee, Dobrowolski and Toruńczyk, this result implies that each closed convex subset of a Fréchet space is homeomorphic to [ 0 , 1 ] × [ 0 , 1 ) m × ( κ ) for some cardinals 0 ≤ n ≤ ω, 0 ≤ m ≤ 1 and κ ≥ 0.

A "hidden" characterization of approximatively polyhedral convex sets in Banach spaces

Taras BanakhIvan Hetman — 2012

Studia Mathematica

A closed convex subset C of a Banach space X is called approximatively polyhedral if for each ε > 0 there is a polyhedral (= intersection of finitely many closed half-spaces) convex set P ⊂ X at Hausdorff distance < ε from C. We characterize approximatively polyhedral convex sets in Banach spaces and apply the characterization to show that a connected component of the space C o n v ( X ) of closed convex subsets of X endowed with the Hausdorff metric is separable if and only if contains a polyhedral convex...

A homological selection theorem implying a division theorem for Q-manifolds

Taras BanakhRobert Cauty — 2007

Banach Center Publications

We prove that a space M with Disjoint Disk Property is a Q-manifold if and only if M × X is a Q-manifold for some C-space X. This implies that the product M × I² of a space M with the disk is a Q-manifold if and only if M × X is a Q-manifold for some C-space X. The proof of these theorems exploits the homological characterization of Q-manifolds due to Daverman and Walsh, combined with the existence of G-stable points in C-spaces. To establish the existence of such points we prove (and afterward...

On universality of countable and weak products of sigma hereditarily disconnected spaces

Taras BanakhRobert Cauty — 2001

Fundamenta Mathematicae

Suppose a metrizable separable space Y is sigma hereditarily disconnected, i.e., it is a countable union of hereditarily disconnected subspaces. We prove that the countable power X ω of any subspace X ⊂ Y is not universal for the class ₂ of absolute G δ σ -sets; moreover, if Y is an absolute F σ δ -set, then X ω contains no closed topological copy of the Nagata space = W(I,ℙ); if Y is an absolute G δ -set, then X ω contains no closed copy of the Smirnov space σ = W(I,0). On the other hand, the countable power X ω of...

The algebraic dimension of linear metric spaces and Baire properties of their hyperspaces.

Taras BanakhAnatolij Plichko — 2006

RACSAM

Answering a question of Halbeisen we prove (by two different methods) that the algebraic dimension of each infinite-dimensional complete linear metric space X equals the size of X. A topological method gives a bit more: the algebraic dimension of a linear metric space X equals |X| provided the hyperspace K(X) of compact subsets of X is a Baire space. Studying the interplay between Baire properties of a linear metric space X and its hyperspace, we construct a hereditarily Baire linear metric space...

Applications of some results of infinite-dimensional topology to the topological classification of operator images

This volume consists of three relatively independent articles devoted to the topological study of the so-called operator images and weak unit balls of Banach spaces. These articles are: “The topological classification of weak unit balls of Banach spaces” by T. Banakh, “The topological and Borel classification of operator images” by T. Banakh, T. Dobrowolski and A. Plichko, and “Operator images homeomorphic to Σ ω ” by T. Banakh. The articles summarize investigations that has been done by these authors...

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