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For a point y in Y, with f(y)=x, the ramification index of f at y is a positive integer e_y such that there is some open neighborhood U of y so that x has only one preimage ...

Let f:D(z_0,r)\{z_0}->C be analytic and bounded on a punctured open disk D(z_0,r), then lim_(z->z_0)f(z) exists, and the function defined by f^~:D(z_0,r)->C f^~(z)={f(z) for ...

Let f be differentiable on the open interval (a,b) and continuous on the closed interval [a,b]. Then if f(a)=f(b), then there is at least one point c in (a,b) where f^'(c)=0. ...

A topological space having a countable dense subset. An example is the Euclidean space R^n with the Euclidean topology, since it has the rational lattice Q^n as a countable ...

A function on the reals R is a step function if it can be written as a finite linear combination of semi-open intervals [a,b) subset= R. Therefore, a step function f can be ...

A continuous group G which has the topology of a T2-space is a topological group. The simplest example is the group of real numbers under addition. The homeomorphism group of ...

A disk with radius 1. The (open) unit disk can also be considered to be the region in the complex plane defined by {z:|z|<1}, where |z| denotes the complex modulus. (The ...

Let gamma(G) denote the domination number of a simple graph G. Then Vizing (1963) conjectured that gamma(G)gamma(H)<=gamma(G×H), where G×H is the graph product. While the ...

A Borel set is an element of a Borel sigma-algebra. Roughly speaking, Borel sets are the sets that can be constructed from open or closed sets by repeatedly taking countable ...

A branch cut is a curve (with ends possibly open, closed, or half-open) in the complex plane across which an analytic multivalued function is discontinuous. For convenience, ...