The strength of compactness for countable complete linear orders

We investigate the statement “the order topology of every countable complete linear order is compact” in the framework of reverse mathematics, and we find that the statement’s strength depends on the precise formulation of compactness. If we require that open covers must be uniformly expressible as unions of basic open sets, then the compactness of complete linear orders is equivalent to WKL0 over RCA0. If open covers need not be uniformly expressible as unions of basic open sets, then the compactness of complete linear orders is equivalent to ACA0 over RCA0. This answers a question of Francois Dorais.