  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 $\mathsf{WKL}_0$ over $\mathsf{RCA}_0$. If open covers need not
be uniformly expressible as unions of basic open sets, then the compactness of
complete linear orders is equivalent to $\mathsf{ACA}_0$ over $\mathsf{RCA}_0$.
This answers a question of Fran\c{c}ois Dorais.
