closeordered

Closeordered is not a widely standardized term in mathematics, but it is used informally to describe structures that combine a linear order with a topology in a way that closeness and order interact coherently. In this sense, closeordered spaces are ordered topological spaces where the order structure and the topology reinforce each other, and where notions of convergence and closure respect the underlying order.

A common informal definition considers an ordered topological space (X, ≤, τ) and imposes a compatibility condition: monotone convergence with respect to the order implies convergence in the topology. More precisely, if a net or sequence is increasing (or decreasing) with respect to ≤ and converges in the order sense to a point x, then it should converge to x in τ. The converse is sometimes required as well, depending on the source. The real line with its standard order and usual topology is often cited as a canonical example, since its order convergence aligns with ordinary topological convergence for monotone sequences.

Variations of the idea may emphasize different compatibility requirements. Some authors require that τ be the order topology generated by the order, or that closed order-intervals [a, b] behave nicely with respect to τ, for instance being τ-closed or τ-compact under certain conditions. Others may focus on continuity of order-theoretic operations (such as suprema or infima) with respect to the topology.

Relationship to other concepts is closeordered spaces: they intersect topics in order topology, lattice-ordered spaces, and domain theory, and are studied to understand how order-theoretic properties interact with topological ones. Because the term is not standardized, readers should verify the exact definition used in a given source. See also order topology, ordered topological space, and lattice-ordered structures.