Čech cohomology of . Every subcover is also a refinement, but the opposite is not always true. A subcover is made from the sets that are in the cover, but omitting some of them; whereas a refinement is made from any sets that are subsets of the sets in the cover. The refinement relation on the set of covers of is transitive, irreflexive, and asymmetric. Generally speaking, a refinement of a given structure is another that in some sense contains it. Examples are to be found when partitioning an interval (one refinement of being ), considering topologies (the standard topology in Euclidean space being a refinement of the trivial topology). When subdividing simplicial complexes (the first barycentric subdivision of a simplicial complex is a refinement), the situation is slightly different: every simplex in the finer complex is a face of some simplex in the coarser one, and both have equal underlying polyhedra. Yet another notion of refinement is that of star refinement. Subcover A simple way to get a subcover is to omit the sets contained in another set in the cover. Consider specifically open covers. Let be a topological basis of and be an open cover of First take Then is a refinement of . Next, for each we select a containing (requiring the axiom of choice). Then is a subcover of Hence the cardinality of a subcover of an open cover can be as small as that of any topological basis. Hence in particular second countability implies a space is Lindelöf. Compactness The language of covers is often used to define several topological properties related to compactness. A topological space X is said to be Compact if every open cover has a finite subcover, (or equivalently that every open cover has a finite refinement); Lindelöf if every open cover has a countable subcover, (or equivalently that every open cover has a countable refinement); Metacompact if every open cover has a point-finite open refinement; Paracompact if every open cover admits a locally finite open refinement. For some more variations see the above articles. Covering dimension A topological space X is said to be of covering dimension n if every open cover of X has a point-finite open refinement such that no point of X is included in more than n+1 sets in the refinement and if n is the minimum value for which this is true. If no such