Extremally disconnected space

• Every discrete space is extremally disconnected. Every indiscrete space is both extremally disconnected and connected. • The Stone–Čech compactification of a discrete space is extremally disconnected. • The spectrum of an abelian von Neumann algebra is extremally disconnected. • Any commutative AW*-algebra is isomorphic to C(X), for some space X which is extremally disconnected, compact and Hausdorff. • Any infinite space with the cofinite topology is both extremally disconnected and connected. More generally, every hyperconnected space is extremally disconnected. • The space on three points with base \{\{x,y\},\{x,y,z\}\} provides a finite example of a space that is both extremally disconnected and connected. Another example is given by the Sierpinski space, since it is finite, connected, and hyperconnected. The following spaces are not extremally disconnected: • The Cantor set is not extremally disconnected. However, it is totally disconnected. ==Equivalent characterizations==

A theorem due to says that the projective objects of the category of compact Hausdorff spaces are exactly the extremally disconnected compact Hausdorff spaces. A simplified proof of this fact is given by . A compact Hausdorff space is extremally disconnected if and only if it is a retract of the Stone–Čech compactification of a discrete space. ==Applications==

proves the Riesz–Markov–Kakutani representation theorem by reducing it to the case of extremally disconnected spaces, in which case the representation theorem can be proved by elementary means. == See also ==