Open subset
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 -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 |
