Every component is a closed subset of the original space.

A closed (in this topology) subset of is called a boundary of if for all .

Let be an open set in and a closed discrete subset.

In fact, S contains a closed subset of order type ω.

In particular, this holds for closed subsets.

A closed subset of a compact space is compact.

A time scale (or measure chain) is a closed subset of the real line .

Conversely, the domain of an induced substructure is a closed subset.

The closed subsets (or induced substructures) of a structure form a lattice.

The join of two subsets is the closed subset generated by their union.