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 |
---|---|
Canonical name | ExtremallyDisconnected |
Date of creation | 2013-03-22 12:42:00 |
Last modified on | 2013-03-22 12:42:00 |
Owner | PrimeFan (13766) |
Last modified by | PrimeFan (13766) |
Numerical id | 8 |
Author | PrimeFan (13766) |
Entry type | Definition |
Classification | msc 54G05 |
Synonym | extremely disconnected |
Related topic | ConnectedSpace |