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# 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.

Related:

ConnectedSpace

Synonym:

extremely disconnected

Type of Math Object:

Definition

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Reference

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## Mathematics Subject Classification

54G05*no label found*

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