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The expression im kleinen is German and means "on a small scale." A topological space is connected im kleinen at a point x if every neighborhood U of x contains an open ...

A topological space that is not connected, i.e., which can be decomposed as the disjoint union of two nonempty open subsets. Equivalently, it can be characterized as a space ...

In real and functional analysis, equicontinuity is a concept which extends the notion of uniform continuity from a single function to collection of functions. Given ...

An equivalence relation on a set X is a subset of X×X, i.e., a collection R of ordered pairs of elements of X, satisfying certain properties. Write "xRy" to mean (x,y) is an ...

Erdős offered a $3000 prize for a proof of the proposition that "If the sum of reciprocals of a set of integers diverges, then that set contains arbitrarily long arithmetic ...

A place nu of a number field k is an isomorphism class of field maps k onto a dense subfield of a nondiscrete locally compact field k_nu. In the function field case, let F be ...

Let G be a group and S be a topological G-set. Then a closed subset F of S is called a fundamental domain of G in S if S is the union of conjugates of F, i.e., S= union _(g ...

A presentation of a group is a description of a set I and a subset R of the free group F(I) generated by I, written <(x_i)_(i in I)|(r)_(r in R)>, where r=1 (the identity ...

Informally, self-similar objects with parameters N and s are described by a power law such as N=s^d, where d=(lnN)/(lns) is the "dimension" of the scaling law, known as the ...

A local ring is a ring R that contains a single maximal ideal. In this case, the Jacobson radical equals this maximal ideal. One property of a local ring R is that the subset ...