Ultraconnected

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]

Properties

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]

Examples

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]

See also

• Hyperconnected space

Notes

References

• 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).