Uniformly connected space
Encyclopedia
In topology
and related areas of mathematics
a uniformly connected space or Cantor connected space is a uniform space
U such that every uniformly continuous function from U to a discrete uniform space is constant.
A uniform space
U is called uniformly disconnected if it is not uniformly connected.
uniform space
is uniformly connected if and only if
it is connected
Topology
Topology is a major area of mathematics concerned with properties that are preserved under continuous deformations of objects, such as deformations that involve stretching, but no tearing or gluing...
and related areas of mathematics
Mathematics
Mathematics is the study of quantity, space, structure, and change. Mathematicians seek out patterns and formulate new conjectures. Mathematicians resolve the truth or falsity of conjectures by mathematical proofs, which are arguments sufficient to convince other mathematicians of their validity...
a uniformly connected space or Cantor connected space is a uniform space
Uniform space
In the mathematical field of topology, a uniform space is a set with a uniform structure. Uniform spaces are topological spaces with additional structure which is used to define uniform properties such as completeness, uniform continuity and uniform convergence.The conceptual difference between...
U such that every uniformly continuous function from U to a discrete uniform space is constant.
A uniform space
Uniform space
In the mathematical field of topology, a uniform space is a set with a uniform structure. Uniform spaces are topological spaces with additional structure which is used to define uniform properties such as completeness, uniform continuity and uniform convergence.The conceptual difference between...
U is called uniformly disconnected if it is not uniformly connected.
Properties
A compactCompact space
In mathematics, specifically general topology and metric topology, a compact space is an abstract mathematical space whose topology has the compactness property, which has many important implications not valid in general spaces...
uniform space
Uniform space
In the mathematical field of topology, a uniform space is a set with a uniform structure. Uniform spaces are topological spaces with additional structure which is used to define uniform properties such as completeness, uniform continuity and uniform convergence.The conceptual difference between...
is uniformly connected if and only if
If and only if
In logic and related fields such as mathematics and philosophy, if and only if is a biconditional logical connective between statements....
it is connected
Connected space
In topology and related branches of mathematics, a connected space is a topological space that cannot be represented as the union of two or more disjoint nonempty open subsets. Connectedness is one of the principal topological properties that is used to distinguish topological spaces...
Examples
- every connected spaceConnected spaceIn topology and related branches of mathematics, a connected space is a topological space that cannot be represented as the union of two or more disjoint nonempty open subsets. Connectedness is one of the principal topological properties that is used to distinguish topological spaces...
is uniformly connected - the rational numberRational numberIn mathematics, a rational number is any number that can be expressed as the quotient or fraction a/b of two integers, with the denominator b not equal to zero. Since b may be equal to 1, every integer is a rational number...
s and the irrational numbers are disconnected but uniformly connected