Journal of Formalized Mathematics
Volume 9, 1997
University of Bialystok
Copyright (c) 1997 Association of Mizar Users

Intermediate Value Theorem and Thickness of Simple Closed Curves


Yatsuka Nakamura
Shinshu University, Nagano
Andrzej Trybulec
University of Bialystok

Summary.

Various types of the intermediate value theorem ( [14]) are proved. For their special cases, the Bolzano theorem is also proved. Using such a theorem, it is shown that if a curve is a simple closed curve, then it is not horizontally degenerated, neither is it vertically degenerated.

MML Identifier: TOPREAL5

The terminology and notation used in this paper have been introduced in the following articles [15] [18] [1] [17] [19] [4] [12] [6] [13] [2] [10] [3] [7] [8] [9] [11] [16] [5]

Contents (PDF format)

  1. Intermediate Value Theorems and Bolzano Theorem
  2. Simple Closed Curves Are Not Flat

Bibliography

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[3] Leszek Borys. Paracompact and metrizable spaces. Journal of Formalized Mathematics, 3, 1991.
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[5] Czeslaw Bylinski and Piotr Rudnicki. Bounding boxes for compact sets in $\calE^2$. Journal of Formalized Mathematics, 9, 1997.
[6] Agata Darmochwal. Compact spaces. Journal of Formalized Mathematics, 1, 1989.
[7] Agata Darmochwal. The Euclidean space. Journal of Formalized Mathematics, 3, 1991.
[8] Agata Darmochwal and Yatsuka Nakamura. Metric spaces as topological spaces --- fundamental concepts. Journal of Formalized Mathematics, 3, 1991.
[9] Agata Darmochwal and Yatsuka Nakamura. The topological space $\calE^2_\rmT$. Simple closed curves. Journal of Formalized Mathematics, 3, 1991.
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[13] Konrad Raczkowski and Pawel Sadowski. Topological properties of subsets in real numbers. Journal of Formalized Mathematics, 2, 1990.
[14] Georgi E. Shilov, editor. \em Elementary Real and Complex Analysis(English translation, translated by Richard A. Silverman). The MIT Press, 1973.
[15] Andrzej Trybulec. Tarski Grothendieck set theory. Journal of Formalized Mathematics, Axiomatics, 1989.
[16] Andrzej Trybulec. A Borsuk theorem on homotopy types. Journal of Formalized Mathematics, 3, 1991.
[17] Andrzej Trybulec. Subsets of real numbers. Journal of Formalized Mathematics, Addenda, 2003.
[18] Zinaida Trybulec. Properties of subsets. Journal of Formalized Mathematics, 1, 1989.
[19] Edmund Woronowicz. Relations and their basic properties. Journal of Formalized Mathematics, 1, 1989.

Received November 13, 1997


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