In
topology
Topology (from the Greek language, Greek words , and ) is the branch of mathematics concerned with the properties of a Mathematical object, geometric object that are preserved under Continuous function, continuous Deformation theory, deformat ...
and related areas of
mathematics
Mathematics is a field of study that discovers and organizes methods, Mathematical theory, theories and theorems that are developed and Mathematical proof, proved for the needs of empirical sciences and mathematics itself. There are many ar ...
a uniformly connected space or Cantor connected space is a
uniform space
In the mathematical field of topology, a uniform space is a topological space, set with additional mathematical structure, structure that is used to define ''uniform property, uniform properties'', such as complete space, completeness, uniform con ...
''U'' such that every
uniformly continuous function from ''U'' to a
discrete uniform space is constant.
A
uniform space
In the mathematical field of topology, a uniform space is a topological space, set with additional mathematical structure, structure that is used to define ''uniform property, uniform properties'', such as complete space, completeness, uniform con ...
''U'' is called uniformly disconnected if it is not uniformly connected.
Properties
A
compact
Compact as used in politics may refer broadly to a pact or treaty; in more specific cases it may refer to:
* Interstate compact, a type of agreement used by U.S. states
* Blood compact, an ancient ritual of the Philippines
* Compact government, a t ...
uniform space
In the mathematical field of topology, a uniform space is a topological space, set with additional mathematical structure, structure that is used to define ''uniform property, uniform properties'', such as complete space, completeness, uniform con ...
is uniformly connected
if and only if
In logic and related fields such as mathematics and philosophy, "if and only if" (often shortened as "iff") is paraphrased by the biconditional, a logical connective between statements. The biconditional is true in two cases, where either bo ...
it is
connected
Examples
* every
connected space
In topology and related branches of mathematics, a connected space is a topological space that cannot be represented as the union (set theory), union of two or more disjoint set, disjoint Empty set, non-empty open (topology), open subsets. Conne ...
is uniformly connected
* the
rational number
In mathematics, a rational number is a number that can be expressed as the quotient or fraction of two integers, a numerator and a non-zero denominator . For example, is a rational number, as is every integer (for example,
The set of all ...
s and the
irrational numbers are disconnected but uniformly connected
See also
*
connectedness
In mathematics, connectedness is used to refer to various properties meaning, in some sense, "all one piece". When a mathematical object has such a property, we say it is connected; otherwise it is disconnected. When a disconnected object can be ...
References
#
Cantor, Georg ''Über Unendliche, lineare punktmannigfaltigkeiten'',
Mathematische Annalen. 21 (1883) 545-591.
Uniform spaces
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