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Concept# Ultraconnected space

Summary

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 \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.
Every ultraconnected space is normal, limit point compact, and pseudocompact.
Examples
The following

Official source

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