Submaximal space: Difference between revisions

Latest revision as of 01:52, 27 January 2012

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

Definition

A topological space is termed submaximal if it satisfies the following equivalent conditions:

Every subset of it is locally closed, i.e., an intersection of an open subset and a closed subset.
Every dense subset is open.
Every preopen subset is open.

Relation with other properties

Stronger properties

Door space

Weaker properties

Weakly submaximal space

@@ Line 3: / Line 3: @@
 ==Definition==
-===Symbol-free definition===
+A [[topological space]] is termed '''submaximal''' if it satisfies the following equivalent conditions:
-A [[topological space]] is termed '''submaximal''' if every subset of it is [[locally closed subset|locally closed]], viz, an intersection of an [[open subset]] and a [[closed subset]].
+# Every subset of it is [[defining ingredient::locally closed subset|locally closed]], i.e., an intersection of an [[defining ingredient::open subset]] and a [[defining ingredient::closed subset]].
+# Every [[defining ingredient::dense subset]] is [[defining ingredient::open subset|open]].
-In particular, this means that every [[dense subset]] is [[open subset|open]].
+# Every [[defining ingredient::preopen subset]] is [[defining ingredient::open subset|open]].
 ==Relation with other properties==