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Ukrainian Mathematical Journal

, Volume 67, Issue 3, pp 431–454 | Cite as

Kronrod–Reeb Graphs of Functions on Noncompact Two-Dimensional Surfaces. I

  • E. A. Polulyakh
Article

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 , 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.

Keywords

Singular Point Topological Space Open Neighborhood Regular Point Continuous Curve 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

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Copyright information

© Springer Science+Business Media New York 2015

Authors and Affiliations

  • E. A. Polulyakh
    • 1
  1. 1.Institute of MathematicsUkrainian National Academy of SciencesKyivUkraine

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