In mathematics

Mathematics

Mathematics is the study of quantity, space, structure, and change. Mathematicians seek out patterns and formulate new conjectures. Mathematicians resolve the truth or falsity of conjectures by mathematical proofs, which are arguments sufficient to convince other mathematicians of their validity...

, a topological space

Topological space

Topological spaces are mathematical structures that allow the formal definition of concepts such as convergence, connectedness, and continuity. They appear in virtually every branch of modern mathematics and are a central unifying notion...

 is termed extremally disconnected or extremely disconnected if the closure of every open set in it is open. (The term "extremally disconnected" is usual, even though the word "extremally" does not appear in most dictionaries.)

An extremally disconnected space that is also compact

Compact space

In mathematics, specifically general topology and metric topology, a compact space is an abstract mathematical space whose topology has the compactness property, which has many important implications not valid in general spaces...

 and Hausdorff

Hausdorff space

In topology and related branches of mathematics, a Hausdorff space, separated space or T2 space is a topological space in which distinct points have disjoint neighbourhoods. Of the many separation axioms that can be imposed on a topological space, the "Hausdorff condition" is the most frequently...

 is sometimes called a Stonean space. (Note that this is different from a Stone space, which is usually a totally disconnected compact Hausdorff space.) A theorem due to Andrew Gleason

Andrew Gleason

Andrew Mattei Gleason was an American mathematician and the eponym of Gleason's theorem and the Greenwood–Gleason graph. After briefly attending Berkeley High School he graduated from Roosevelt High School in Yonkers, then Yale University in 1942, where he became a Putnam Fellow...

 says that the projective objects of the category of compact Hausdorff spaces are exactly the extremally disconnected compact Hausdorff spaces. Just as there is a duality between Stone spaces and Boolean algebras, there is a duality between Stonean spaces and the category of complete Boolean algebra

Complete Boolean algebra

In mathematics, a complete Boolean algebra is a Boolean algebra in which every subset has a supremum . Complete Boolean algebras are used to construct Boolean-valued models of set theory in the theory of forcing...

s.

An extremally disconnected first countable collectionwise Hausdorff space

Collectionwise Hausdorff space

In mathematics, in the field of topology, a topological space is said to be collectionwise Hausdorff if given any closed discrete collection of points in the topological space, there are pairwise disjoint open sets containing the points...

 must be discrete. In particular, for metric space

Metric space

In mathematics, a metric space is a set where a notion of distance between elements of the set is defined.The metric space which most closely corresponds to our intuitive understanding of space is the 3-dimensional Euclidean space...

s, 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).

Examples

• Every discrete space
  Discrete space
  In topology, a discrete space is a particularly simple example of a topological space or similar structure, one in which the points are "isolated" from each other in a certain sense.- Definitions :Given a set X:...
  
   is extremally disconnected.
• The Stone–Čech compactification
  Stone–Cech compactification
  In the mathematical discipline of general topology, Stone–Čech compactification is a technique for constructing a universal map from a topological space X to a compact Hausdorff space βX...
  
   of a discrete space is extremally disconnected.
• The spectrum of an abelian von Neumann algebra
  Abelian von Neumann algebra
  In functional analysis, an abelian von Neumann algebra is a von Neumann algebra of operators on a Hilbert space in which all elements commute.The prototypical example of an abelian von Neumann algebra is...
  
  is extremally disconnected.