Kronrod–Reeb Graphs of Functions on Noncompact Two-Dimensional Surfaces. I
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We consider continuous functions on two-dimensional surfaces satisfying the following conditions: they have a discrete set of local extrema; if a point is not a local extremum, then there exist its neighborhood and a number n ∈ ℕ such that a 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 the level sets of f. It is known that, for compact M 2 , the space Γ K−R (f) is a topological graph. We introduce the notion of graph with stalks, which generalizes the notion of topological graph. For noncompact M 2 , we establish three conditions sufficient for Γ K−R (f) to be a graph with stalks.
KeywordsSingular Point Topological Space Open Neighborhood Regular Point Continuous Curve
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