Extremally disconnected space

In mathematics, an extremally disconnected space is a topological space in which the closure of every open set is open.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.The following spaces are not extremally disconnected: A theorem due to Gleason (1958) says that the projective objects of the category of compact Hausdorff spaces are exactly the extremally disconnected compact Hausdorff spaces.[2] Hartig (1983) 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.
topological spacehomophonecompactHausdorffStone spacetotally disconnectedBoolean algebrascomplete Boolean algebrasfirst-countablecollectionwise Hausdorff spacediscretemetric spacesdiscrete spaceindiscrete spaceStone–Čech compactificationspectrumabelian von Neumann algebraAW*-algebracofinite topologyconnectedhyperconnected spacefiniteSierpinski spaceCantor setprojective objectscategoryretractRiesz–Markov–Kakutani representation theoremTotally disconnected spaceOxford English DictionaryOxford University PressA. V. ArkhangelskiiEncyclopedia of MathematicsEMS PressGleason, Andrew M.American Mathematical MonthlyJohnstone, Peter T.Rainwater, John