Closed 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

This article is about a basic definition in topology.
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Definition

A subset of a topological space is termed closed if it satisfies the following equivalent conditions:

• Its set-theoretic complement is an open subset
• It contains all its limit points

Relation with other properties

Stronger properties

• Clopen subset

Weaker properties

• Locally closed subset