Open subset: Difference between revisions

Latest revision as of 00:27, 27 January 2012

This article defines a property over pairs of a topological space and a subspace, or equivalently, properties over subspace embeddings (viz, subsets) in topological spaces

A subset of a topological space has this property in the space iff its set-theoretic complement in the whole space is a/an: closed subset

This article is about a basic definition in topology.
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Definition

A subset of a topological space is termed open if it satisfies the following equivalent conditions:

In terms of the standard definition of topology in terms of open subsets: It is one of the member of the topology
In terms of a basis: It is a union (possibly empty) of basis open sets
In terms of a subbasis: It is a union (possibly empty) of finite intersections of subbasis open sets
In terms of closed subsets: It is the set-theoretic complement of a closed subset

Relation with other properties

Stronger properties

Property	Meaning	Proof of implication	Proof of strictness (reverse implication failure)	Intermediate notions
regular open subset	interior of its closure	(direct from definition)	open not implies regular open	Template:Intermediate notion short
open dense subset

Weaker properties

Property	Meaning	Proof of implication	Proof of strictness (reverse implication failure)	Intermediate notions
G-delta subset (denoted $G_{\delta }$ -subset)	countable intersection of open subsets	open implies G-delta	G-delta not implies open	\|FULL LIST, MORE INFO
preopen subset	contained in the interior of its closure	open implies preopen	preopen not implies open	\|FULL LIST, MORE INFO
semiopen subset	contained in the closure of its interior	open implies semiopen	semiopen not implies open	\|FULL LIST, MORE INFO

@@ Line 9: / Line 9: @@
 A subset of a [[topological space]] is termed '''open''' if it satisfies the following equivalent conditions:
-* In terms of the standard definition of topology: It is one of the member of the ''topology''
+* In terms of the standard definition of topology in terms of open subsets: It is one of the member of the ''topology''
 * In terms of a [[basis for a topological space|basis]]: It is a union (possibly empty) of basis open sets
 * In terms of a [[subbasis for a topological space|subbasis]]: It is a union (possibly empty) of finite intersections of subbasis open sets
 * In terms of [[closed subset]]s: It is the set-theoretic complement of a closed subset
+==Relation with other properties==
+===Stronger properties===
+{| class="sortable" border="1"
+! Property !! Meaning !! Proof of implication !! Proof of strictness (reverse implication failure) !! Intermediate notions
+|-
+| [[Weaker than::regular open subset]] || [[interior]] of its [[closure]] || (direct from definition) || [[open not implies regular open]] || {{intermediate notion short|open subset|regular open subset}}
+|-
+| [[Weaker than::open dense subset]] || || || ||
+|}
+===Weaker properties===
+{| class="sortable" border="1"
+! Property !! Meaning !! Proof of implication !! Proof of strictness (reverse implication failure) !! Intermediate notions
+|-
+| [[Stronger than::G-delta subset]] (denoted <math>G_\delta</math>-subset) || countable intersection of open subsets || [[open implies G-delta]] || [[G-delta not implies open]] || {{intermediate notions short|G-delta subset|open subset}}
+|-
+| [[Stronger than::preopen subset]] || contained in the interior of its closure || [[open implies preopen]] || [[preopen not implies open]] || {{intermediate notions short|preopen subset|open subset}}
+|-
+| [[Stronger than::semiopen subset]] || contained in the closure of its interior || [[open implies semiopen]] || [[semiopen not implies open]] || {{intermediate notions short|semiopen subset|open subset}}
+|}