| Popis: |
This chapter presents a collection of problems formulated by participants and guests of the Lviv topological seminar held at the Ivan Franko Lviv National University in Ukraine. The chapter begins discussion on asymptotic dimension by recalling that a metric space X is proper if the distance d (·, x 0 ) to a fixed point is a proper map for any x 0 ∈ X . A map f : X → Y between metric spaces is called coarse if it satisfies the following two conditions: (1) Coarse Uniformity: There is a monotone function λ: [0,∞) → [0,∞) such that d Y ( f ( x ), f ( x 1 )) ≤ λ ( dX ( x , x 1 )); (2) Metric Properness: The preimage f − 1 ( B ) is bounded for every bounded set B ⊂ Y . Two maps f , g into a metric space Y are close if there exists a constant C > 0 such that d Y ( f ( x ), g ( x )) C , for every C > 0 . Two metric spaces X , Y are said to be coarse equivalent if there exist coarse maps f : X → Y and g : Y → X such that the maps g f and 1 X are close and also f g and 1 Y are close. This chapter also explains concepts related to extension of metrics, questions in general topology, and some problems in Ramsey theory. |
