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Let X be a Riemann surface.Then the intersection number of two closed curves on X has a simple definition in terms of an integral. For every closed curve c on X (i.e., smooth function :), we can associate a differential form of compact support, the Poincaré dual of c, with the property that integrals along c can be calculated by integrals over X:
Any intersection of finitely many elements of τ is an element of τ. If τ is a topology on X, then the pair (X, τ) is called a topological space. The notation X τ may be used to denote a set X endowed with the particular topology τ. By definition, every topology is a π-system. The members of τ are called open sets in X.
In mathematics, intersection theory is one of the main branches of algebraic geometry, where it gives information about the intersection of two subvarieties of a given variety. [1] The theory for varieties is older, with roots in Bézout's theorem on curves and elimination theory. On the other hand, the topological theory more quickly reached a ...
The intersection of any finite number of elements of F is again in F. If A is in F and if B contains A, then B is in F. Final topology On a set X with respect to a family of functions into , is the finest topology on X which makes those functions continuous. [9] Fine topology (potential theory)
Let be a set and a nonempty family of subsets of ; that is, is a subset of the power set of . Then is said to have the finite intersection property if every nonempty finite subfamily has nonempty intersection; it is said to have the strong finite intersection property if that intersection is always infinite.
The covering dimension is the smallest number n such that for every cover, there is a refinement in which every point in X lies in the intersection of no more than n + 1 covering sets. This is the gist of the formal definition below.
An important example for number theory is the group p of p-adic integers, for a prime number p, meaning the inverse limit of the finite groups /p n as n goes to infinity. The group Z {\displaystyle \mathbb {Z} } p is well behaved in that it is compact (in fact, homeomorphic to the Cantor set ), but it differs from (real) Lie groups in that it ...
In topology, the degree of a continuous mapping between two compact oriented manifolds of the same dimension is a number that represents the number of times that the domain manifold wraps around the range manifold under the mapping. The degree is always an integer, but may be positive or negative depending on the orientations.