about complete metric spaces, and its consequences, such as the open mapping theorem and the closed graph theorem. On every infinite-dimensional topological vector space there is a discontinuous linear map. General topology A uniform space is compact if and only if it is complete and totally bounded. Every Tychonoff space has a Stone–Čech compactification. Mathematical logic Gödel's completeness theorem for first-order logic: every consistent set of first-order sentences has a completion. That is, every consistent set of first-order sentences can be extended to a maximal consistent set. The compactness theorem: If is a set of first-order (or alternatively, zero-order) sentences such that every finite subset of has a model, then has a model. Possibly equivalent implications of AC There are several historically important set-theoretic statements implied by AC whose equivalence to AC is open. The partition principle, which was formulated before AC itself, was cited by Zermelo as a justification for believing AC. In 1906, Russell declared PP to be equivalent, but whether the partition principle implies AC is still the oldest open problem in set theory, and the equivalences of the other statements are similarly hard old open problems. In every known model of ZF where choice fails, these statements fail too, but it is unknown if they can hold without choice. Set theory Partition principle: if there is a surjection from A to B, there is an injection from B to A. Equivalently, every partition P of a set S is less than or equal to S in size. Converse Schröder–Bernstein theorem: if two sets have surjections to each other, they are equinumerous. Weak partition principle: if there is a injection and a surjection from A to B, then A and B are equinumerous. Equivalently, a partition of a set S cannot be strictly larger than S. If WPP holds, this already implies the existence of a non-measurable set. Each of the previous three statements is implied by the preceding one, but it is unknown if any of these implications can be reversed. There is no infinite decreasing sequence of cardinals. The equivalence was conjectured by Schoenflies in 1905. Abstract algebra Hahn embedding theorem: Every ordered abelian group G order-embeds as a subgroup of the additive group endowed with a lexicographical order, where Ω is the set of Archimedean equivalence classes of G. This equivalence was conjectured by Hahn in 1907. Stronger forms of the negation of AC