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

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: 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