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
In mathematics, a real function f of real numbers is said to be uniformly continuous if there is a positive real number \delta such that function values over any function domain interval of the size \delta are as close to each other as we want. In ...
from ''U'' to a
discrete uniform space is constant.
A
uniform space ''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
* Blood compact, an ancient ritual of the Philippines
* Compact government, a type of colonial rule utilized in Britis ...
uniform space is uniformly connected
if and only if it is
connected
Connected may refer to:
Film and television
* ''Connected'' (2008 film), a Hong Kong remake of the American movie ''Cellular''
* '' Connected: An Autoblogography About Love, Death & Technology'', a 2011 documentary film
* ''Connected'' (2015 TV ...
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 ...
is uniformly connected
* the
rational numbers 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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