# Open subset

From Topospaces

*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.VIEW: Definitions built on this | Facts about this | Survey articles about this

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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 -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 |