Erdos space and homeomorphism groups of manifolds

Abstract

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...

Characterizing stable complete Erdős space

“…Proof. The first statement follows immediately from the case ν = 1 which was proved in [12,Lemma 4.14]. We verify the part about Lelek functions for the case ν = ω.…”

“…The proof of the uniqueness of the Lelek fan and of Lelek functions is based on this fact; see [5], [6], and [12,Theorem 6.2]. In this section we will prove the uniqueness of ω-Lelek functions.…”

“…The topologies of E and E c are characterized in [10,12] and [13], respectively. Both spaces are universal elements of the class of almost zero-dimensional spaces; see [12,Theorem 4.15]. It is proved in [12,Corollary 9.4] that E is homeomorphic to…”

“…Both spaces are universal elements of the class of almost zero-dimensional spaces; see [12,Theorem 4.15]. It is proved in [12,Corollary 9.4] that E is homeomorphic to…”

“…A cohesive space is obviously at least one-dimensional at every point but the converse is not valid; see Dijkstra [8]. We will use the following result from Dijkstra and van Mill [12,Lemma 5.9] about the connection between cohesion and Lelek functions. …”