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 |
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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 |