extremally disconnected
A topological space
is said to be extremally disconnected if every open set in has an open closure
.
It can be shown that is extremally disconnected iff any two disjoint open sets in 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 |