On the Subcontinua of a Real Line

Adam Grabowski

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

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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