Preopen 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
Definition
A subset of a topological space is termed a preopen subset if it satisfies the following equivalent conditions:
- It is contained in the interior of its closure.
- It is the intersection of a regular open subset and a dense subset.
- It is the intersection of an open subset and a dense subset.
Equivalence of definitions
Further information: equivalence of definitions of preopen subset
Relation with other properties
Stronger properties
| Property | Meaning | Proof of implication | Proof of strictness (reverse implication failure) | Intermediate notions |
|---|---|---|---|---|
| open subset | open implies preopen | follows from existence of dense subsets that are not open | |FULL LIST, MORE INFO | |
| regular open subset | equals the interior of its closure | |||
| dense subset | ||||
| open dense subset |