Properties of topological spaces

Extremally disconnected space

In mathematics, an extremally disconnected space is a topological space in which the closure of every open set is open. (The term "extremally disconnected" is correct, even though the word "extremally" does not appear in most dictionaries, and is sometimes mistaken by spellcheckers for the homophone extremely disconnected.) An extremally disconnected space that is also compact and Hausdorff is sometimes called a Stonean space. This is not the same as a Stone space, which is a totally disconnected compact Hausdorff space. Every Stonean space is a Stone space, but not vice versa. In the duality between Stone spaces and Boolean algebras, the Stonean spaces correspond to the complete Boolean algebras. An extremally disconnected first-countable collectionwise Hausdorff space must be discrete. In particular, for metric spaces, the property of being extremally disconnected (the closure of every open set is open) is equivalent to the property of being discrete (every set is open). (Wikipedia).

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Related pages

Topological space | Metric space | Totally disconnected space | Riesz–Markov–Kakutani representation theorem | Complete Boolean algebra | Finite set | Stone–Čech compactification | Hyperconnected space | Abelian von Neumann algebra | Stone space | First-countable space | Base (topology) | Projective object | Hausdorff space | AW*-algebra | Connected space | Spectrum (functional analysis) | Collectionwise Hausdorff space | Category (mathematics) | Compact space | Discrete space | Boolean algebra (structure)