# extremally disconnected

A topological space $X$ is said to be extremally disconnected if every open set in $X$ has an open closure.

It can be shown that $X$ is extremally disconnected iff any two disjoint open sets in $X$ have disjoint closures. Every extremally disconnected space is totally disconnected.

## Notes

Some authors like [1] and [2] use the above definition as is, while others (e.g. [3, 4]) require that an extremally disconnected space should (in addition to the above condition) also be a Hausdorff space.

## References

• 1 S. Willard, General Topology, Addison-Wesley, Publishing Company, 1970.
• 2 J. L. Kelley, General Topology, D. van Nostrand Company, Inc., 1955.
• 3 L. A. Steen, J. A. Seebach, Jr., Counterexamples in topology, Holt, Rinehart and Winston, Inc., 1970.
• 4 N. Bourbaki, General Topology, Part 1, Addison-Wesley Publishing Company, 1966.
Title extremally disconnected ExtremallyDisconnected 2013-03-22 12:42:00 2013-03-22 12:42:00 PrimeFan (13766) PrimeFan (13766) 8 PrimeFan (13766) Definition msc 54G05 extremely disconnected ConnectedSpace