Clopen subset

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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 clopen subset' of a topological space is a subset that is both open and closed.

Facts

The empty subspace, and the whole space, are always clopen subsets.

In a connected space, these are the only clopen subsets. In general, the clopen subsets occur as the unions of connected components of the topological space.