scholarly journals UNIVERSAL MINIMAL FLOWS OF GENERALIZED WAŻEWSKI DENDRITES

2018 ◽  
Vol 83 (04) ◽  
pp. 1618-1632 ◽  
Author(s):  
ALEKSANDRA KWIATKOWSKA
Keyword(s):  
Minimal Flow ◽  
Topological Groups ◽  
Homeomorphism Group ◽  
Image Position ◽  
Homeomorphism Groups

AbstractWe study universal minimal flows of the homeomorphism groups of generalized Ważewski dendrites WP, $P \subseteq \left\{ {3,4, \ldots ,\omega } \right\}$. If P is finite, we prove that the universal minimal flow of the homeomorphism group H (WP) is metrizable and we compute it explicitly. This answers a question of Duchesne. If P is infinite, we show that the universal minimal flow of H (WP) is not metrizable. This provides examples of topological groups which are Roelcke precompact and have a nonmetrizable universal minimal flow with a comeager orbit.

scholarly journals Long Colimits of Topological Groups III: Homeomorphisms of Products and Coproducts

Axioms ◽  
2021 ◽  
Vol 10 (3) ◽  
pp. 155
Author(s):  
Rafael Dahmen ◽  
Gábor Lukács
Keyword(s):  
Compact Support ◽  
Necessary Conditions ◽  
Sufficient Conditions ◽  
Topological Groups ◽  
Tychonoff Space ◽  
Compactly Supported ◽  
Homeomorphism Group ◽  
Finite Products ◽  
Homeomorphism Groups ◽  
Necessary And Sufficient

The group of compactly supported homeomorphisms on a Tychonoff space can be topologized in a number of ways, including as a colimit of homeomorphism groups with a given compact support or as a subgroup of the homeomorphism group of its Stone-Čech compactification. A space is said to have the Compactly Supported Homeomorphism Property (CSHP) if these two topologies coincide. The authors provide necessary and sufficient conditions for finite products of ordinals equipped with the order topology to have CSHP. In addition, necessary conditions are presented for finite products and coproducts of spaces to have CSHP.

scholarly journals The universal minimal flow of the homeomorphism group of the Lelek fan

2018 ◽  
Vol 371 (10) ◽  
pp. 6995-7027 ◽  
Author(s):  
Dana Bartošová ◽  
Aleksandra Kwiatkowska
Keyword(s):  
Minimal Flow ◽  
Homeomorphism Group

scholarly journals A Theorem on Algebras of Measures on Topological Groups

1959 ◽  
Vol 11 (4) ◽  
pp. 195-206 ◽  
Author(s):  
J. H. Williamson
Keyword(s):  
Banach Space ◽  
Abelian Group ◽  
Compact Abelian Group ◽  
Locally Compact Abelian Group ◽  
Topological Groups ◽  
Finite Sets ◽  
Locally Compact ◽  
Borel Measures ◽  
Bounded Complex ◽  
Image Position

Let G be a locally compact Abelian group, and the set of bounded complex (regular countably-additive Borel) measures on G. It is well known that becomes a Banach space if the norm is defined bythe supremum being over all finite sets of disjoint Borel subsets of G.

scholarly journals LECs, Local Mixers, Topological Groups and Special Products

1988 ◽  
Vol 44 (2) ◽  
pp. 252-258 ◽  
Author(s):  
Carlos R. Borges
Keyword(s):  
Topological Group ◽  
Topological Groups ◽  
Symmetric Products ◽  
Free Topological Group ◽  
Reduced Products ◽  
Contractible Spaces ◽  
Homeomorphism Groups

AbstractWe prove that every (locally) contractible topological group is (L)EC and apply these results to homeomorphism groups, free topological groups, reduced products and symmetric products. Our main results are: The free topological group of a θ-contractible space is equiconnected. A paracompact and weakly locally contractible space is locally equiconnected if and only if it has a local mixer. There exist compact metric contractible spaces X whose reduced (symmetric) products are not retracts of the Graev free topological groups F(X) (A(X)) (thus correcting results we published ibidem).

scholarly journals Two structure theorems for homeomorphism groups

1971 ◽  
Vol 4 (1) ◽  
pp. 63-68
Author(s):  
A. R. Vobach
Keyword(s):  
Metric Spaces ◽  
Present Note ◽  
Topological Properties ◽  
Compact Metric Space ◽  
Natural Structure ◽  
Compact Metric Spaces ◽  
Continuous Map ◽  
Structure Theorems ◽  
Image Position ◽  
Homeomorphism Groups

Let H(C) be the group of homeomorphisms of the cantor set, C onto itself. Let p: C → M be a (continuous) map of C onto a compact metric space M, and let G(p, M) be {h ∈ H(C) | ∀x ∈ C, p(x) = ph(x)}. G(p, M) is a group. The map p: C → M is standard, if for each (x, y) ∈ C × C such that p(x) = p(y), there is a sequence and a sequence such that xn → x and hn(xn) → y. Standard maps and their associated groups characterize compact metric spaces in the sense that: Two such spaces, M and N, are homeomorphic if and only if, given p standard from C onto M, there is a standard q from C onto N for which G(p, M) = h−1G(q, N)h, for some h ∈ H(C). That is, two compact metric spaces are homeomorphic if and only if they determine, via standard maps, the same classes of conjugate subgroups of H(C).The present note exhibits two natural structure theorems relating algebraic and topological properties: First, if M = H ∪ K (H, K ≠ π) , compact metric, and p : C → M are given, then G(p, M) is isomorphic to a subdirect product of G(p, M)/S(p, H\K) and G(p, M)/S(p, K\H) where, generally, S(p, N) is the normal subgroup of homeomorphisms supported on p−1M . Second, given M and N compact metric and p : M → N continuous and onto, let M ≠ M − CID*α ≠ 0 , where {Dα}α ∈ A is the collection of non-degenerate preimages of points in N Then there is a standard p : C → M such that fp : C → N is standard and there is a homomorphism.

Lifting amalgamated sums and other colimits of groups and topological groups

1987 ◽  
Vol 102 (2) ◽  
pp. 273-280 ◽  
Author(s):  
Ronald Brown ◽  
Philip R. Heath
Keyword(s):  
Free Product ◽  
Topological Groups ◽  
Free Product With Amalgamation ◽  
Image Position

Suppose a group H is given as a free product with amalgamationdetermined by groups A0, A1, A2 and homomorphisms α1: A0 → A1, α2: A0 → A2. Thus H may be described as the quotient of the free product A * A2 by the relations i1 α1 (α0) = i2α2 (α0) for all α0 ∈ A0, where i1, i2 are the two injections of A1, A2 into A1 * A2. We do not assume that α1, α2 are injective, so the canonical homomorphisms α′i: Ai → H, i = 0,1,2, also need not be injective.

scholarly journals UNIVERSAL SUBGROUPS OF POLISH GROUPS

2014 ◽  
Vol 79 (4) ◽  
pp. 1148-1183 ◽  
Author(s):  
KONSTANTINOS A. BEROS
Keyword(s):  
Topological Group ◽  
Banach Spaces ◽  
Topological Groups ◽  
Locally Compact ◽  
Polish Groups ◽  
Countable Power ◽  
Compact Case ◽  
Image Position ◽  
The Relationship ◽  
Compactly Generated

AbstractGiven a class${\cal C}$of subgroups of a topological groupG, we say that a subgroup$H \in {\cal C}$is auniversal${\cal C}$subgroupofGif every subgroup$K \in {\cal C}$is a continuous homomorphic preimage ofH. Such subgroups may be regarded as complete members of${\cal C}$with respect to a natural preorder on the set of subgroups ofG. We show that for any locally compact Polish groupG, the countable powerGωhas a universalKσsubgroup and a universal compactly generated subgroup. We prove a weaker version of this in the nonlocally compact case and provide an example showing that this result cannot readily be improved. Additionally, we show that many standard Banach spaces (viewed as additive topological groups) have universalKσand compactly generated subgroups. As an aside, we explore the relationship between the classes ofKσand compactly generated subgroups and give conditions under which the two coincide.

scholarly journals A note an homeomorphic measures on topological groups

1983 ◽  
Vol 26 (2) ◽  
pp. 169-171
Author(s):  
Sidney A. Morris ◽  
Vincent C. Peck
Keyword(s):  
Probability Measures ◽  
Hilbert Cube ◽  
Dimensional Manifold ◽  
Topological Groups ◽  
Von Neumann ◽  
Finite Dimensional ◽  
Measurable Set ◽  
Ulam Theorem ◽  
Borel Probability ◽  
Image Position

The classical von Neumann–Oxtoby–Ulam Theorem states the following:Given non-atomic Borel probability measures μ, λ on In such thatthere exists a homeomorphism h of In onto itself fixing the boundary pointwise such that for any λ-measurable set SIt is known that the above theorem remains valid if In is replaced by any compact finite dimensional manifold [2], [4] or with I∞, the Hilbert cube, [8].

SEMILATTICES AND THE RAMSEY PROPERTY

2015 ◽  
Vol 80 (4) ◽  
pp. 1236-1259 ◽  
Author(s):  
MIODRAG SOKIĆ
Keyword(s):  
Automorphism Group ◽  
Minimal Flow ◽  
Linear Extensions ◽  
Ramsey Property ◽  
Linear Orderings ◽  
Topological Interpretation ◽  
Finite Lattices ◽  
Image Position ◽  
Fraïssé Class ◽  
Uniquely Ergodic

AbstractWe consider${\cal S}$, the class of finite semilattices;${\cal T}$, the class of finite treeable semilattices; and${{\cal T}_m}$, the subclass of${\cal T}$which contains trees with branching bounded bym. We prove that${\cal E}{\cal S}$, the class of finite lattices with linear extensions, is a Ramsey class. We calculate Ramsey degrees for structures in${\cal S}$,${\cal T}$, and${{\cal T}_m}$. In addition to this we give a topological interpretation of our results and we apply our result to canonization of linear orderings on finite semilattices. In particular, we give an example of a Fraïssé class${\cal K}$which is not a Hrushovski class, and for which the automorphism group of the Fraïssé limit of${\cal K}$is not extremely amenable (with the infinite universal minimal flow) but is uniquely ergodic.

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