Extremally disconnected space: Difference between revisions
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A [[topological space]] is said to be '''extremally disconnected''' if it satisfies the following equivalent conditions: | A [[topological space]] is said to be '''extremally disconnected''' if it satisfies the following equivalent conditions: | ||
# Any [[regular open subset]] (i.e., the interior of any [[closed subset]) is closed. | # Any [[regular open subset]] (i.e., the interior of any [[closed subset]]) is closed. | ||
# The closure of any open subset is open. | # The closure of any open subset is open. | ||
# The intersection of two [[semiopen subset]]s is semiopen. | # The intersection of two [[semiopen subset]]s is semiopen. | ||
Latest revision as of 02:11, 27 January 2012
Definition
Symbol-free definition
A topological space is said to be extremally disconnected if it satisfies the following equivalent conditions:
- Any regular open subset (i.e., the interior of any closed subset) is closed.
- The closure of any open subset is open.
- The intersection of two semiopen subsets is semiopen.
- The semiopen subsets form a topology, i.e., they are closed under taking finite intersections and arbitrary unions.
This article defines a property of topological spaces: a property that can be evaluated to true/false for any topological space|View a complete list of properties of topological spaces
This is an opposite of connectedness