Extremally disconnected space: Difference between revisions
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# 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. | ||
# The semiopen subsets form a | # The semiopen subsets form a topology, i.e., they are closed under taking finite intersections and arbitrary unions. | ||
{{topospace property}} | {{topospace property}} | ||
{{oppositeof|connectedness}} | {{oppositeof|connectedness}} | ||
==Relation with other properties== | ==Relation with other properties== | ||
Revision as of 01:44, 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