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Brunnian links

100%
Gartside P., Greenwood S.
Fundamenta Mathematicae
|
2007
|
tom 193
|
nr 3
259-276
EN
A Brunnian link is a set of n linked loops such that every proper sublink is trivial. Simple Brunnian links have a natural algebraic representation. This is used to determine the form, length and number of minimal simple Brunnian links. Braids are used to investigate when two algebraic words represent equivalent simple Brunnian links that differ only in the arrangement of the component loops.
2
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Function spaces and local properties

100%
Feng Z., Gartside P.
Fundamenta Mathematicae
|
2013
|
tom 223
|
nr 3
207-223
EN
Necessary conditions and sufficient conditions are given for $C_p(X)$ to be a (σ-) m₁- or m₃-space. (A space is an m₁-space if each of its points has a closure-preserving local base.) A compact uncountable space K is given with $C_{p}(K)$ an m₁-space, which answers questions raised by Dow, Ramírez Martínez and Tkachuk (2010) and Tkachuk (2011).
3
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Calibres, compacta and diagonals

100%
Gartside P., Morgan J.
Fundamenta Mathematicae
|
2016
|
tom 232
|
nr 1
1-19
EN
For a space Z let 𝒦(Z) denote the partially ordered set of all compact subspaces of Z under set inclusion. If X is a compact space, Δ is the diagonal in X², and 𝒦(X²∖Δ) has calibre (ω₁,ω), then X is metrizable. There is a compact space X such that X²∖Δ has relative calibre (ω₁,ω) in 𝒦(X²∖Δ), but which is not metrizable. Questions of Cascales et al. (2011) concerning order constraints on 𝒦(A) for every subspace of a space X are answered.
4
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The complexity of the set of squares in the homeomorphism group of the circle

100%
Gartside P., Pejić B.
Fundamenta Mathematicae
|
2007
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tom 195
|
nr 2
125-134
EN
The set of squares in the group of autohomeomorphisms of the circle is complete analytic, and hence analytic but not Borel.
5
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n-Arc connected spaces

81%
Espinoza B., Gartside P., Mamatelashvili A.
Colloquium Mathematicum
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2013
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tom 130
|
nr 2
221-240
EN
A space is n-arc connected (n-ac) if any family of no more than n-points are contained in an arc. For graphs the following are equivalent: (i) 7-ac, (ii) n-ac for all n, (iii) continuous injective image of a closed subinterval of the real line, and (iv) one of a finite family of graphs. General continua that are ℵ₀-ac are characterized. The complexity of characterizing n-ac graphs for n = 2,3,4,5 is determined to be strictly higher than that of the stated characterization of 7-ac graphs.
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