all q-simplices the cocycle condition holds. A 0-cocycle is a collection of local sections of satisfying a compatibility relation on every intersecting A 1-cocycle satisfies for every non-empty with Coboundary A q-cochain is called a q-coboundary if it is in the image of and is the set of all q-coboundaries. For example, a 1-cochain is a 1-coboundary if there exists a 0-cochain such that for every intersecting Cohomology The Čech cohomology of with values in is defined to be the cohomology of the cochain complex . Thus the qth Čech cohomology is given by . The Čech cohomology of X is defined by considering refinements of open covers. If is a refinement of then there is a map in cohomology The open covers of X form a directed set under refinement, so the above map leads to a direct system of abelian groups. The Čech cohomology of X with values in is defined as the direct limit of this system. The Čech cohomology of X with coefficients in a fixed abelian group A, denoted , is defined as where is the constant sheaf on X determined by A. A variant of Čech cohomology, called numerable Čech cohomology, is defined as above, except that all open covers considered are required to be numerable: that is, there is a partition of unity {ρi} such that each support is contained in some element of the cover. If X is paracompact and Hausdorff, then numerable Čech cohomology agrees with the usual Čech cohomology. Relation to other cohomology theories If X is homotopy equivalent to a CW complex, then the Čech cohomology is naturally isomorphic to the singular cohomology . If X is a differentiable manifold, then is also naturally isomorphic to the de Rham cohomology; the article on de Rham cohomology provides a brief review of this isomorphism. For less well-behaved spaces, Čech cohomology differs from singular cohomology. For example if X is the closed topologist's sine curve, then whereas If X is a differentiable manifold and the cover of X is a "good cover" (i.e. all the sets Uα are contractible to a point, and all finite intersections of sets in are either empty or contractible to a point), then is isomorphic to the de Rham cohomology. If X is compact Hausdorff, then Čech cohomology (with coefficients in a discrete group) is isomorphic to Alexander-Spanier cohomology. For a presheaf on X, let denote