non trivial intersection. Hence, no T1 space with more than one point is ultraconnected. Every ultraconnected space X {\displaystyle X} is path-connected...

space is ultraconnected if no two non-empty closed sets are disjoint. Every ultraconnected space is path-connected. Indiscrete or trivial. A space is indiscrete...

components. Every Noetherian topological space has finitely many irreducible components. Ultraconnected space Sober space Geometrically irreducible Steen & Seebach...

whole space X. For example, the particular point topology on a finite space is hyperconnected while the excluded point topology is ultraconnected. The...

is constant. The Sierpiński space S is both hyperconnected (since every nonempty open set contains 1) and ultraconnected (since every nonempty closed...

and thus X is not ultraconnected. Note that if X is the Sierpiński space then no such a and b exist and X is in fact ultraconnected. Compact only if finite...

X is connected if and only if A(X) is directly indecomposable X is ultraconnected if and only if A(X) is finitely subdirectly irreducible X is compact...

{\displaystyle P} has a single point, X ∗ {\displaystyle X^{*}} is ultraconnected. For a set Z and a point p in Z, one obtains the excluded point topology...

neighborhoods. The space is ultraconnected, as any nonempty closed set contains the point p . {\displaystyle p.} Therefore the space is also connected...

X {\displaystyle X} is a Baire space. X {\displaystyle X} is second-countable. X {\displaystyle X} is ultraconnected, since the closures of the singletons...