In
topology and related areas of
mathematics
Mathematics is an area of knowledge that includes the topics of numbers, formulas and related structures, shapes and the spaces in which they are contained, and quantities and their changes. These topics are represented in modern 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
In topology, a discrete space is a particularly simple example of a topological space or similar structure, one in which the points form a , meaning they are ''isolated'' from each other in a certain sense. The discrete topology is the finest t ...
is constant.
A
uniform space ''U'' is called uniformly disconnected if it is not uniformly connected.
Properties
A
compact uniform space is uniformly connected
if and only if 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 of two or more disjoint non-empty open subsets. Connectedness is one of the principal topological properties tha ...
is uniformly connected
* the
rational numbers and the
irrational numbers are disconnected but uniformly connected
See also
*
connectedness
References
#
Cantor, Georg ''Über Unendliche, lineare punktmannigfaltigkeiten'',
Mathematische Annalen
''Mathematische Annalen'' (abbreviated as ''Math. Ann.'' or, formerly, ''Math. Annal.'') is a German mathematical research journal founded in 1868 by Alfred Clebsch and Carl Neumann. Subsequent managing editors were Felix Klein, David Hilbert, ...
. 21 (1883) 545-591.
Uniform spaces
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