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extremally disconnected
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(Definition)
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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.
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.
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"extremally disconnected" is owned by PrimeFan. [ full author list (3) | owner history (2) ]
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Cross-references: Hausdorff space, addition, totally disconnected, disjoint, iff, closure, 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 3072 times total.
Classification:
| AMS MSC: | 54G05 (General topology :: Peculiar spaces :: Extremally disconnected spaces, $F$-spaces, etc.) |
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Pending Errata and Addenda
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