|
|
|
|
extremally disconnected
|
(Definition)
|
|
|
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.
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.
- 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.
|
"extremally disconnected" is owned by PrimeFan. [ full author list (3) | owner history (2) ]
|
|
(view preamble)
Cross-references: Hausdorff space, addition, totally disconnected, disjoint, iff, closure, open, open set, topological space
There are 2 references to this entry.
This is version 5 of extremally disconnected, born on 2002-06-01, modified 2008-03-19.
Object id is 2982, canonical name is ExtremallyDisconnected.
Accessed 2248 times total.
Classification:
| AMS MSC: | 54G05 (General topology :: Peculiar spaces :: Extremally disconnected spaces, $F$-spaces, etc.) |
|
|
|
|
|
|
Pending Errata and Addenda
|
|
|
|
|
|
|
|
|
|
|
