Heine-Borel Theorem Closed and bounded in Rn is equivalent to compact. Proof: Assume E is compact, which means that every open cover of it has a finite subcover. Apply this to any arbitrary covering of E by rational open balls, and since a finite subcover exists, the subcover is bounde...
In the short study, we present a contradiction which comes from ignoring a hidden assumption in Heine-Borel theorem. Finally, we prove a useful theorem which rectifies the contradiction. 2010 Mathematics subject Classification: 30L99 Keywords and Phrases: Metric Space, Compactness, Heine-Borel ...
We put a metric on the space of infinite binary sequences and prove that compactness of this space follows from a simple combinatorial lemma. The Heine-Borel theorem is an immediate corollary.doi:10.48550/arXiv.0808.0844Macauley, MatthewRabern, Brian...