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 abovedefinition as is, while others (e.g. [3, 4]) requirethat an extremally disconnected space should (in addition to the abovecondition) 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.

单词	ExtremallyDisconnected
释义	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 abovedefinition as is, while others (e.g. [3, 4]) requirethat an extremally disconnected space should (in addition to the abovecondition) 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.
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