On one-point connectifications

Abry, Mohammad; Dijkstra, J.; Mill, Jan van

doi:10.1016/j.topol.2006.09.004

Cited by 9 publications

(32 citation statements)

References 15 publications

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“…Since by the definition Y t ⊂ Y s whenever t ∈ succ(s) we have that T is a tree and we also have condition (1) for the system (Y s ) s∈T . Condition (6) follows from hypothesis (a) and conditions (4) and (3) follow from hypothesis (b) and, for (3), compactness. We verify that for every s ∈ A <ω ,…”

Section: Note That Hypothesis (B) Is Satisfied Because Y S Is Clopen mentioning

confidence: 95%

“…We shall see that U works for condition (2 ) as well as (4 ) (2 ) is satisfied. Now let C be a nonempty clopen subset of some E s such that C ⊂ U .…”

mentioning

confidence: 87%

“…It is shown by Abry, Dijkstra, and van Mill [4] that there are totally disconnected cohesive spaces that do not admit a one-point connectification.…”

Section: Jan J Dijkstra and Jan Van Millmentioning

confidence: 99%

“…Condition (4). Let x ∈ E and let U be a neighbourhood of x such that U contains no nonempty clopen subsets of any E t with the ρ-topology.…”

Section: Intrinsic Characterizations Of Erdős Space 41mentioning

confidence: 99%

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Erdos space and homeomorphism groups of manifolds

Dijkstra

Mill

2010

Memoirs of the AMS

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Subscription renewals are subject to late fees. See www.ams.org/help-faq for more journal subscription information. Each number may be ordered separately; please specify number when ordering an individual number.Back number information. For back issues see www.ams.org/bookstore. Subscriptions and orders should be addressed to the American Mathematical Society, P. O. Box 845904, Boston, MA 02284-5904 USA. All orders must be accompanied by payment. Other correspondence should be addressed to 201 Charles Street, Providence, RI 02904-2294 USA.Copying and reprinting. Individual readers of this publication, and nonprofit libraries acting for them, are permitted to make fair use of the material, such as to copy a chapter for use in teaching or research. Permission is granted to quote brief passages from this publication in reviews, provided the customary acknowledgment of the source is given.Republication, systematic copying, or multiple reproduction of any material in this publication is permitted only under license from the American Mathematical Society. Requests for such permission should be addressed to the Acquisitions Department, American Mathematical Society, 201 Charles Street, Providence, Rhode Island 02904-2294 USA. Requests can also be made by e-mail to reprint-permission@ams.org. iii Memoirs of the American Mathematical Society AbstractLet M be either a topological manifold, a Hilbert cube manifold, or a Menger manifold and let D be an arbitrary countable dense subset of M . Consider the topological group H(M, D) which consists of all autohomeomorphisms of M that map D onto itself equipped with the compact-open topology. We present a complete solution to the topological classification problem for H(M, D) as follows. If M is a one-dimensional topological manifold, then we proved in an earlier paper that H(M, D) is homeomorphic to Q ω , the countable power of the space of rational numbers. In all other cases we find in this paper that H(M, D) is homeomorphic to the famed Erdős space E, which consists of the vectors in Hilbert space 2 with rational coordinates. We obtain the second result by developing topological characterizations of Erdős space. If X is compact then the standard topology on the group of homeomorphisms H(X) of X is the so-called compact-open topology (which coincides with the topology of uniform convergence). For noncompact locally compact spaces we give H(X) the topology that this group inherits from H(αX), where αX is the one-point compactification. In either case we have that H(X) a Polish topological group. If A is a subset of a space X then H(X, A) stands for the subgroup {h ∈ H(X) : h(A) = A} of H(X).Brouwer [11] showed that R is countable dense homogeneous, that is, for all countable dense subsets A and B of R there is an h ∈ H(R) with h(A) = B. It is not difficult to prove that every R n has this property. In view of Brouwer's result it is a natural idea to investigate the group H(R n , Q n ). It was shown in Dijkstra and van Mill [21] that the group H(R, Q) is homeomorphic to the z...

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Section: Note That Hypothesis (B) Is Satisfied Because Y S Is Clopen mentioning

confidence: 95%

“…We shall see that U works for condition (2 ) as well as (4 ) (2 ) is satisfied. Now let C be a nonempty clopen subset of some E s such that C ⊂ U .…”

mentioning

confidence: 87%

“…It is shown by Abry, Dijkstra, and van Mill [4] that there are totally disconnected cohesive spaces that do not admit a one-point connectification.…”

Section: Jan J Dijkstra and Jan Van Millmentioning

confidence: 99%

“…Condition (4). Let x ∈ E and let U be a neighbourhood of x such that U contains no nonempty clopen subsets of any E t with the ρ-topology.…”

Section: Intrinsic Characterizations Of Erdős Space 41mentioning

confidence: 99%

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Erdos space and homeomorphism groups of manifolds

Dijkstra

Mill

2010

Memoirs of the AMS

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“…Theorem 1.2 deals with the space I = Exh(ϕ) = Fin(ϕ). First we generalize the definitions of Exh(ϕ) and Fin(ϕ) as given in (1) to the function χ :…”

Section: Claim 35 ϕ Satisfies the Properties (A) Through (D)mentioning

confidence: 99%

On generalized Erdős spaces

Dijkstra

Visser

2008

Topology and its Applications

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During the last years both Erdős space and complete Erdős space were topologically characterized by Dijkstra and van Mill. Applications include results about Erdős type spaces in p-spaces as well as results about Polishable ideals on ω. We present an unifying theorem in terms of sets with a reflexive relation that among other things contains these apparently dissimilar results as special cases.

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