Can I determine the finite subcover of any given open cover?(An unknown side of a known result)

Question

Heine-Borel theorem in $\mathbb R$ states that for a closed,bounded set,we can get a finite subcover for every given open cover.But the proof is existential and uses completeness(in the form of Cantor-intersection theorem).My question is given a closed bounded set in $\mathbb R$ and any particular open cover,is it possible to construct a finite subcover ,I mean is it possible to understand which one of the given ones should I take in the finite subcollection so that it is also a cover.For an easy open cover of a closed bounded interval,it is easy by observation to find a finite one,but is it possible for any given open cover,is it easy or even possible to determine which ones are essential as covers.I doubt it is not possible,otherwise Heine Borel would have given a constructive rather than existential proof and things would become very easy. Can we somehow visualize the selection process of finite subcover or we have just existence in hand? Is this some equivalent version of Markov axiom in logic theory?

Answer 1

The proof of the Heine–Borel theorem may not be explicitly presented in the form of a construction, but it suggests a construction:

Without loss of generality, let the bounded, closed set $S$ be a subset of $(0,1)$. Given an open cover $\mathcal C$, first add the open set $(0,1)\setminus S$ to $\mathcal C$. For each $k\in\mathbb N$, check whether each $\left[j\cdot2^{-k},(j+1)\cdot2^{-k}\right]$ with $j=0,1,\ldots,2^k-1$ is covered by some element of the cover. If so, these $2^k$ elements form a finite subcover. The process ends at some finite $k$, at the latest when each of the non-empty pairwise overlaps of the elements of the existing finite open subcover contains at least one of the points $j\cdot2^{-k}$.

Now remove $(0,1)\setminus S$ from the finite cover if necessary.