Open subset

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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