A Study on Nice Open Covers in Constructive Analysis

TL;DR

This paper investigates conditions under which certain open covers of the interval [0,1] in constructive analysis admit finite sub-covers, contrasting classical and constructive perspectives on compactness.

Contribution

It introduces specific restrictions on open covers that ensure the existence of finite sub-covers in constructive analysis, extending classical results.

Findings

Certain restricted open covers of [0,1] are constructively compact.

Constructive real numbers between 0 and 1 are not generally compact.

Finite sub-covers can be chosen under specific conditions in constructive setting.

Abstract

Mathematicians like Markov and Bishop made an effort to develop constructive mathematics and extended many theorems in classical mathematical analysis. Heine Borel theorem tells us that a closed bounded subset of Euclidean space R is compact, but in constructive mathematics, Tseitin and Zaslavskii showed that the set of all constructive real numbers between 0 and 1 is not compact. We are going to show that when giving certain restriction to the open cover on [0,1], we can however always choose a finite sub-cover.