Kronrod–Reeb Graphs of Functions on Noncompact Two-Dimensional Surfaces. II
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We consider continuous functions on two-dimensional surfaces satisfying the following conditions: they have a discrete set of local extrema and if a point is not a local extremum, then there exist its neighborhood and a number n ∈ ℕ such that the function restricted to this neighborhood is topologically conjugate to Re z n in a certain neighborhood of zero. Given f : M 2 → ℝ, let Γ K−R (f) be a quotient space of M 2 with respect to its partition formed by the components of level sets of the function f. It is known that the space Γ K−R (f) is a topological graph if M 2 is compact. In the first part of the paper, we introduced the notion of graph with stalks that generalizes the notion of topological graph. For noncompact M 2 , we present three conditions sufficient for Γ K−R (f) to be a graph with stalks. In the second part, we prove that these conditions are also necessary in the case M 2 = ℝ2 . In the general case, one of our conditions is not necessary. We provide an appropriate example.
KeywordsSingular Point Regular Point Local Extremum Hausdorff Space Topological Graph
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