On topological Kadec norms

Abry, Mohammad; Dijkstra, J.

doi:10.1007/s00208-005-0651-5

Cited by 9 publications

(15 citation statements)

References 9 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…Almost zero-dimensionality is clearly hereditary and preserved under products. Oversteegen and Tymchatyn [23] have shown that every almost zero-dimensional space is at most one-dimensional; see also [21,1].…”

Section: Definition 21mentioning

confidence: 99%

Characterizing stable complete Erdős space

Dijkstra

2011

Isr. J. Math.

Self Cite

View full text Add to dashboard Cite

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.

show abstract

Section: Definition 21mentioning

confidence: 99%

Characterizing stable complete Erdős space

Dijkstra

2011

Isr. J. Math.

Self Cite

View full text Add to dashboard Cite

show abstract

“…] n is an (n + 1)-dimensional space (see [10]) that is clearly almost n-dimensional. Since E c (n) contains a copy of E c × [0, 1] n and is contained in N ω n × R we have dim E c (n) = n + 1. If X is almost n-dimensional, then there exists a weaker topology W that is at most n-dimensional.…”

Section: Corollary 7 Every Almost N-dimensional Space Has An Almost mentioning

confidence: 99%

“…Observe that our definition of almost n-dimensionality differs from the extension of almost zero-dimensionality that is featured in Levin and Tymchatyn [14], which they show to be equivalent to regular dimension. A real-valued function ϕ on a topological space X is called lower semi-continuous (LSC ) if {x ∈ X : ϕ(x) > t} is open for each t ∈ R. In [1] the authors present the following characterization theorem.…”

mentioning

confidence: 99%

Universal spaces for almost $n$-dimensionality

Abry

Dijkstra

2007

Proc. Amer. Math. Soc.

Self Cite

View full text Add to dashboard Cite

Abstract. We find universal functions for the class of lower semi-continuous (LSC) functions with at most n-dimensional domain. In an earlier paper we proved that a space is almost n-dimensional if and only if it is homeomorphic to the graph of an LSC function with an at most n-dimensional domain. We conclude that the class of almost n-dimensional spaces contains universal elements (that are topologically complete). These universal spaces can be thought of as higher-dimensional analogues of complete Erdős space.Unless stated otherwise all topologies in this note are assumed to be separable and metrizable.Let n be a nonnegative integer and let X be a topological space. We say that X is almost n-dimensional if there exists a topology W on X such that dim(X, W) ≤ n, W is weaker than the given topology on X, and every point of X has a neighbourhood basis in X consisting of sets that are closed in (X, W). If we substitute n = 0, then we get the notion of an almost zero-dimensional space, which concept has been introduced by Oversteegen and Tymchatyn [16]; see also [13] Both spaces were introduced and shown to be one-dimensional by Paul Erdős

show abstract

“…A topological space is called almost zero-dimensional if every point has a neighbourhood basis consisting of C-sets. Every almost zero-dimensional space is at most one-dimensional; see [11,10,1].An additive subgroup of a vector space is called line-free if it does not contain nontrivial linear subspaces. It is remarked in [2] that a topological classification of the line-free closed subgroups of Banach spaces produces a classification of all closed subgroups of Banach spaces.…”

mentioning

confidence: 99%

mentioning

confidence: 99%

A counterexample concerning line-free groups and complete Erdos space

Dijkstra¹,

Mill²

2006

Proc. Amer. Math. Soc.

Self Cite

View full text Add to dashboard Cite

Abstract. We present a weakly closed, one-dimensional, line-free subgroup of the separable Banach space c that is not homeomorphic to complete Erdős space. The existence of this example disproves a conjecture of Dobrowolski, Grabowski, and Kawamura.Complete Erdős space was first featured by Erdős in [8], who proved that it is totally disconnected and one-dimensional. It can be represented by, for instance,where 2 is the Hilbert space of square summable real sequences. E c is a universal element of the class of almost zero-dimensional spaces; for background information see [11,9,3,4,5]. A subset of a topological space is called a C-set if it can be written as an intersection of clopen subsets of the space. A topological space is called almost zero-dimensional if every point has a neighbourhood basis consisting of C-sets. Every almost zero-dimensional space is at most one-dimensional; see [11,10,1].An additive subgroup of a vector space is called line-free if it does not contain nontrivial linear subspaces. It is remarked in [2] that a topological classification of the line-free closed subgroups of Banach spaces produces a classification of all closed subgroups of Banach spaces. Let G be an arbitrary nondiscrete, weakly closed, line-free, additive subgroup of a separable Banach space E. Dobrowolski, Grabowski, and Kawamura [7] proved that G is homeomorphic to complete Erdős space whenever E is reflexive. In addition, Ancel, Dobrowolski, and Grabowski [2] showed that E contains zero-dimensional examples of such groups G precisely if E contains an isomorphic copy of c 0 . These results prompted Dobrowolski, Grabowski, and Kawamura [7] to formulate the following Conjecture. Every separable, nondiscrete, weakly closed, one-dimensional, linefree subgroup of a Banach space is homeomorphic to E c .We present a counterexample to this conjecture, thereby finding a new topological type that closed subgroups of Banach spaces can have. We shall distinguish our example from E c by the following property of E c . A topological space is called somewhere zero-dimensional if it contains a point at which the space is

show abstract

On topological Kadec norms

Cited by 9 publications

References 9 publications

Characterizing stable complete Erdős space

Characterizing stable complete Erdős space

Universal spaces for almost $n$-dimensionality

A counterexample concerning line-free groups and complete Erdos space

Contact Info

Product

Resources

About