Ultraconnected space

In mathematics, a topological space is said to be ultraconnected if no two nonempty closed sets are disjoint. 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.

Properties

Every ultraconnected space $X$ is path-connected (but not necessarily arc connected). If $a$ and $b$ are two points of $X$ and $p$ is a point in the intersection $\operatorname {cl} \{a\}\cap \operatorname {cl} \{b\}$, the function $f:[0,1]\to X$ defined by $f(t)=a$ if $0\leq t<1/2$, $f(1/2)=p$ and $f(t)=b$ if $1/2<t\leq 1$, is a continuous path between $a$ and $b$.

Every ultraconnected space is normal, limit point compact, and pseudocompact.

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.

See also

• Hyperconnected space

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).