Topological space in which the closure of every open set is open
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,[1] and is sometimes mistaken by spellcheckers for the
homophoneextremely 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-countablecollectionwise 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).
The space on three points with
base 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
Gleason (1958) 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
Rainwater (1959).
A compact Hausdorff space is extremally disconnected if and only if it is a
retract of the Stone–Čech compactification of a discrete space.[2]