Ultraconnected space explained

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 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 \le t < 1 / 2 , f( 1 / 2 ) = p and f(t) = b if 1 / 2 < t \le 1 , is a continuous path between a and 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

References

• • Lynn Arthur Steen and J. Arthur Seebach, Jr., Counterexamples in Topology. Springer-Verlag, New York, 1978. Reprinted by Dover Publications, New York, 1995. (Dover edition).

Notes and References

1. PlanetMath
2. Steen & Seebach, Sect. 4, pp. 29-30
3. Steen & Seebach, example #50, p. 74