Journal of Formalized Mathematics
Volume 15, 2003
University of Bialystok
Copyright (c) 2003 Association of Mizar Users

## On the Subcontinua of a Real Line

University of Bialystok

### Summary.

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.

#### MML Identifier: BORSUK_5

The terminology and notation used in this paper have been introduced in the following articles [23] [27] [2] [24] [22] [25] [28] [4] [5] [26] [19] [7] [21] [14] [17] [18] [1] [9] [6] [10] [15] [8] [20] [16] [13] [12] [3]

#### Contents (PDF format)

1. Preliminaries
2. Intervals
3. Rational and Irrational Numbers
4. Topological Properties of Intervals
5. Subcontinua of a Real Line
6. Sets with Proper Subsets Only

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