Tietze-type theorems on monotone increasing sets
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- by Jean Chan Stanek PDF
- Proc. Amer. Math. Soc. 54 (1976), 286-290 Request permission
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
- V. L. Klee Jr., Convex sets in linear spaces, Duke Math. J. 18 (1951), 443–466. MR 44014 J. Hutchison, Subconvex sets, Dissertation, University of California, Los Angeles, 1970.
- Heinrich Tietze, Über Konvexheit im kleinen und im großen und über gewisse den Punkten einer Menge zugeordnete Dimensionszahlen, Math. Z. 28 (1928), no. 1, 697–707 (German). MR 1544985, DOI 10.1007/BF01181191
- Frederick A. Valentine, Convex sets, McGraw-Hill Series in Higher Mathematics, McGraw-Hill Book Co., New York-Toronto-London, 1964. MR 0170264
Additional Information
- © Copyright 1976 American Mathematical Society
- Journal: Proc. Amer. Math. Soc. 54 (1976), 286-290
- DOI: https://doi.org/10.1090/S0002-9939-1976-0388244-5
- MathSciNet review: 0388244