The Vitali and Heine-Borel covering theorems are house-hold names of analysis, and rightly well-studied in Reverse Mathematics. As shown in Simpson’s excellent monograph (1), for countable coverings of the unit interval, the Heine-Borel theorem is equivalent to WKL (weak Koenig’s lemma), while the Vitali covering theorem is equivalent to WWKL (weak weak Koenig’s lemma). The theorem numbers in (1) are IV.1.2 and X.1.13.
My question is then as follows:
Is there a natural statement X such that (WWKL +X ) $leftrightarrow$ WKL, say over RCA$_0$?
Here, $X$ should be weaker than WKL, obviously. Results in related frameworks (computability theory, Weihrauch reducibility, constructive math, …) are also welcome.
PS: I am asking this question because in the case of uncountable coverings, such an X does exist.
(1) Stephen G. Simpson, Subsystems of second order arithmetic, 2nd ed., Perspectives in Logic, Cambridge University Press, 2009.