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

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.


Singular Point Topological Space Open Neighborhood Regular Point Continuous Curve 
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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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