Volume 15, 2003

University of Bialystok

Copyright (c) 2003 Association of Mizar Users

**Adam Grabowski**- University of Bialystok

- In [11] we showed that the only proper subcontinua of the simple closed curve are arcs and single points. In this article we prove that the only proper subcontinua of the real line are closed intervals. We introduce some auxiliary notions such as $\rbrack a,b\lbrack_{\Bbb Q}$, $\rbrack a,b\lbrack_{\Bbb I\Bbb Q}$ - intervals consisting of rational and irrational numbers respectively. We show also some basic topological properties of intervals.

This work has been partially supported by CALCULEMUS grant HPRN-CT-2000-00102.

- Preliminaries
- Intervals
- Rational and Irrational Numbers
- Topological Properties of Intervals
- Subcontinua of a Real Line
- Sets with Proper Subsets Only

- [1]
Grzegorz Bancerek.
Cardinal numbers.
*Journal of Formalized Mathematics*, 1, 1989. - [2]
Grzegorz Bancerek.
The ordinal numbers.
*Journal of Formalized Mathematics*, 1, 1989. - [3]
Leszek Borys.
Paracompact and metrizable spaces.
*Journal of Formalized Mathematics*, 3, 1991. - [4]
Czeslaw Bylinski.
Functions and their basic properties.
*Journal of Formalized Mathematics*, 1, 1989. - [5]
Czeslaw Bylinski.
Functions from a set to a set.
*Journal of Formalized Mathematics*, 1, 1989. - [6]
Czeslaw Bylinski and Piotr Rudnicki.
Bounding boxes for compact sets in $\calE^2$.
*Journal of Formalized Mathematics*, 9, 1997. - [7]
Agata Darmochwal.
Compact spaces.
*Journal of Formalized Mathematics*, 1, 1989. - [8]
Agata Darmochwal.
Finite sets.
*Journal of Formalized Mathematics*, 1, 1989. - [9]
Agata Darmochwal and Yatsuka Nakamura.
Metric spaces as topological spaces --- fundamental concepts.
*Journal of Formalized Mathematics*, 3, 1991. - [10]
Noboru Endou and Artur Kornilowicz.
The definition of the Riemann definite integral and some related lemmas.
*Journal of Formalized Mathematics*, 11, 1999. - [11]
Adam Grabowski.
On the decompositions of intervals and simple closed curves.
*Journal of Formalized Mathematics*, 14, 2002. - [12]
Stanislawa Kanas, Adam Lecko, and Mariusz Startek.
Metric spaces.
*Journal of Formalized Mathematics*, 2, 1990. - [13]
Andrzej Kondracki.
Basic properties of rational numbers.
*Journal of Formalized Mathematics*, 2, 1990. - [14]
Jaroslaw Kotowicz.
Convergent real sequences. Upper and lower bound of sets of real numbers.
*Journal of Formalized Mathematics*, 1, 1989. - [15]
Jaroslaw Kotowicz.
The limit of a real function at infinity.
*Journal of Formalized Mathematics*, 2, 1990. - [16]
Wojciech Leonczuk and Krzysztof Prazmowski.
Incidence projective spaces.
*Journal of Formalized Mathematics*, 2, 1990. - [17]
Yatsuka Nakamura.
Half open intervals in real numbers.
*Journal of Formalized Mathematics*, 14, 2002. - [18]
Beata Padlewska.
Connected spaces.
*Journal of Formalized Mathematics*, 1, 1989. - [19]
Beata Padlewska and Agata Darmochwal.
Topological spaces and continuous functions.
*Journal of Formalized Mathematics*, 1, 1989. - [20]
Jan Popiolek.
Some properties of functions modul and signum.
*Journal of Formalized Mathematics*, 1, 1989. - [21]
Konrad Raczkowski and Pawel Sadowski.
Topological properties of subsets in real numbers.
*Journal of Formalized Mathematics*, 2, 1990. - [22]
Andrzej Trybulec.
Enumerated sets.
*Journal of Formalized Mathematics*, 1, 1989. - [23]
Andrzej Trybulec.
Tarski Grothendieck set theory.
*Journal of Formalized Mathematics*, Axiomatics, 1989. - [24]
Andrzej Trybulec.
Subsets of real numbers.
*Journal of Formalized Mathematics*, Addenda, 2003. - [25]
Andrzej Trybulec and Czeslaw Bylinski.
Some properties of real numbers operations: min, max, square, and square root.
*Journal of Formalized Mathematics*, 1, 1989. - [26]
Michal J. Trybulec.
Integers.
*Journal of Formalized Mathematics*, 2, 1990. - [27]
Zinaida Trybulec.
Properties of subsets.
*Journal of Formalized Mathematics*, 1, 1989. - [28]
Edmund Woronowicz.
Relations and their basic properties.
*Journal of Formalized Mathematics*, 1, 1989.

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