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Transfinite Asymptotic Dimension
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Asymptotic property C for metric spaces was introduced by Dranishnikos as generalization of finite asymptotic dimension - asdim. It turns out that this property can be viewed as transfinite extension of asymptotic dimension. The original definition was given by Radul. We introduce three equivalent definitions, show that asymptotic property C is closed under products (open problem stated "Open problems in topology II") and prove some other facts, i.e. by defining dimension of a family of metric spaces. Some examples of spaces enjoying countable trasfinite asymptotic dimension are given. We also formulate open problems and state "omega conjecture", which inspired most part of this paper.
Forward citations
Cited by 2 Pith papers
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APD profiles and transfinite asymptotic dimension
An infinity-pseudometric space has transfinite asymptotic dimension at most omega+n exactly when it admits a two-term integral APD profile of width n+1.
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A metric space with its transfinite asymptotic dimension omega + 1
A concrete metric space is constructed whose transfinite and complementary-finite asymptotic dimensions are both omega+1, giving a counterexample to the omega conjecture.
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