On plane dendroids and their end points in the classical sense

Lelek, A.

doi:10.4064/fm-49-3-301-319

Cited by 62 publications

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“…The following facts can be found in Lelek [20]. Lelek functions with compact domain C exist and C must be homeomorphic to the Cantor set.…”

Section: Remark 27mentioning

confidence: 99%

Characterizing stable complete Erdős space

Dijkstra

2011

Isr. J. Math.

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We focus on the space E ω c , the countable infinite power of complete Erdős space Ec. Both spaces are universal spaces for the class of almost zerodimensional spaces. We prove that E ω c has the property that it is stable under multiplication with any complete almost zero-dimensional space. We obtain this result as a corollary to topological characterization theorems that we develop for E ω c . We also show that σ-compacta are negligible in E ω c and that the space is countable dense homogeneous.

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“…The following facts can be found in Lelek [20]. Lelek functions with compact domain C exist and C must be homeomorphic to the Cantor set.…”

Section: Remark 27mentioning

confidence: 99%

Characterizing stable complete Erdős space

Dijkstra

2011

Isr. J. Math.

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“…It was shown in [33] that the endpoint set of the Lelek fan has a one-point connectification. The relevance of this concept to complete Erdős space was recognized in [31], where it was proved that the endpoint set of the Lelek fan is homeomorphic to E c .…”

Section: Jan J Dijkstra and Jan Van Millmentioning

confidence: 99%

“…Chapter 4 is devoted to semi-continuous functions and in particular to Lelek functions. The standard example of a Lelek function is an arclength function for a Lelek fan [33]. These functions are central to the understanding and characterization of Erdős spaces because of the proof by Kawamura, Oversteegen, and Tymchatyn [31] that complete Erdős space is homeomorphic to the endpoint set of the Lelek fan.…”

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confidence: 99%

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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“…We call ϕ a Lelek function if G ϕ is dense in L ϕ . If ϕ is a Lelek function, then the quotient space L ϕ /∞ we obtain when we identify the set C × {∞} to a point in L ϕ is called a Lelek fan (see [11]). According to Bula and Oversteegen [1] and Charatonik [2], the Lelek fans, and consequently also their endpoint sets G ϕ , are topologically unique.…”

Section: Corollary 2 If 0 Is a Cluster Point Of Lim Sup N→∞ E N Thmentioning

confidence: 99%

A Criterion for Erdős Spaces

Dijkstra¹

2005

Proceedings of the Edinburgh Mathematical Society

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In 1940 Paul Erdős introduced the 'rational Hilbert space', which consists of all vectors in the real Hilbert space 2 that have only rational coordinates. He showed that this space has topological dimension one, yet it is totally disconnected and homeomorphic to its square. In this note we generalize the construction of this peculiar space and we consider all subspaces E of the Banach spaces p that are constructed as 'products' of zero-dimensional subsets En of R. We present an easily applied criterion for deciding whether a general space of this type is one dimensional. As an application we find that if such an E is closed in p , then it is homeomorphic to complete Erdős space if and only if dim E > 0 and every En is zero dimensional. The topology on p is generated by the norm LetR be the compactification [−∞, ∞] of R. We extend the p-norm overRN by putting z = ∞ for each z ∈R N \ p . For the remainder of this note let E 1 , E 2 , . . . be a fixed sequence of subsets of R and let E = {z ∈ p : z n ∈ E n for every n ∈ N} be a corresponding subspace of some fixed p . If we choose p = 2 and E n = Q for every n, then E is called Erdős space E and, if E n = R \ Q, then we obtain complete Erdős space E c (cf. [9] and [10]). Erdős [9] proved that Erdős space and complete Erdős space have topological dimension one. We present an easily applied criterion for deciding whether a general space of the type E is one dimensional. As an application we find that if E is closed in p , then it is homeomorphic to complete Erdős space if and only if dim E > 0 and every E n is zero dimensional. Other applications are particularly simple models of complete Erdős space and the Lelek fan. 595

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On plane dendroids and their end points in the classical sense

Cited by 62 publications

References 0 publications

Characterizing stable complete Erdős space

Characterizing stable complete Erdős space

Erdos space and homeomorphism groups of manifolds

A Criterion for Erdős Spaces

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