Ultraconnected space (original) (raw)
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Property of topological spaces
In mathematics, a topological space is said to be ultraconnected if no two nonempty closed sets are disjoint.[1] Equivalently, a space is ultraconnected if and only if the closures of two distinct points always have non trivial intersection. Hence, no T1 space with more than one point is ultraconnected.[2]
Every ultraconnected space X {\displaystyle X} is path-connected (but not necessarily arc connected). If a {\displaystyle a}
and b {\displaystyle b}
are two points of X {\displaystyle X}
and p {\displaystyle p}
is a point in the intersection cl { a } ∩ cl { b } {\displaystyle \operatorname {cl} \{a\}\cap \operatorname {cl} \{b\}}
, the function f : [ 0 , 1 ] → X {\displaystyle f:[0,1]\to X}
defined by f ( t ) = a {\displaystyle f(t)=a}
if 0 ≤ t < 1 / 2 {\displaystyle 0\leq t<1/2}
, f ( 1 / 2 ) = p {\displaystyle f(1/2)=p}
and f ( t ) = b {\displaystyle f(t)=b}
if 1 / 2 < t ≤ 1 {\displaystyle 1/2<t\leq 1}
, is a continuous path between a {\displaystyle a}
and b {\displaystyle b}
.[2]
Every ultraconnected space is normal, limit point compact, and pseudocompact.[1]
The following are examples of ultraconnected topological spaces.
- A set with the indiscrete topology.
- The Sierpiński space.
- A set with the excluded point topology.
- The right order topology on the real line.[3]
- Hyperconnected space
- This article incorporates material from Ultraconnected space on PlanetMath, which is licensed under the Creative Commons Attribution/Share-Alike License.
- Lynn Arthur Steen and J. Arthur Seebach, Jr., Counterexamples in Topology. Springer-Verlag, New York, 1978. Reprinted by Dover Publications, New York, 1995. ISBN 0-486-68735-X (Dover edition).