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Proceedings of the American Mathematical Society

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Tietze-type theorems on monotone increasing sets

Author: Jean Chan Stanek
Journal: Proc. Amer. Math. Soc. 54 (1976), 286-290
MathSciNet review: 0388244
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Abstract | References | Additional Information

Abstract: The Tietze theorem on convex sets is generalized to monotone increasing sets and strictly monotone increasing sets, which include convex sets as a special case. The main theorem is that a closed connected set in $ {E_2}$ is monotone increasing if and only if it is locally monotone increasing. A similar result is proved for strictly monotone increasing sets.

References [Enhancements On Off] (What's this?)

  • [1] V. L. Klee, Jr., Convex sets in linear spaces, Duke Math. J. 18 (1951), 443-466. MR 13, 354. MR 0044014 (13:354f)
  • [2] J. Hutchison, Subconvex sets, Dissertation, University of California, Los Angeles, 1970.
  • [3] H. Tietze, Über Konvexheit im kleinen und im grossen und über gewisse den Punkten einer Menge zugeordnete Dimensionszahlen, Math. Z. 28 (1928), 697-707. MR 1544985
  • [4] F. A. Valentine, Convex sets, McGraw-Hill Ser. in Higher Math., McGraw-Hill, New York, 1964. MR 30 #503. MR 0170264 (30:503)

Additional Information

Keywords: Convex sets, Tietze's theorem, local convexity, monotone increasing sets, strictly monotone increasing sets
Article copyright: © Copyright 1976 American Mathematical Society

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