The closure of `S` is variously denoted by "Cl(`S`)" or "". If there is more than one topology on `X` (say **T** and **T**'), then the different topologies may give rise to different closures; this can be indicated in the notation by a subscript, as in "Cl_{T}(`S`)".
If the topology is itself defined by some other structure, such as a metric `d`, then "`d`" can be placed in the subscript instead of "**T**".

In a metric space `X` (such as the `n`-dimensional Euclidean space) the closure Cl(`S`) is the set {`x` ∈ `X` : `d`(`x`,`S`) = 0} of all points in `X` whose distance from `S` is 0.
Here, `d`(`x`,`S`) is defined as the infimum of the set {`d`(`x`,`y`) : `y` ∈ `S`}.

In a first countable space (such as a metric space), Cl(`S`) is the set of all limits of all convergent sequences of points in `S`.
For a general topological space, this statement remains true if one replaces "sequence" by "net".

Another characterization of Cl(`S`) is as follows: an element `x` of `X` belongs to Cl(`S`) if and only if every neighborhood of `x` contains an element of `S`.
In other words, `x` ∈ Cl(`S`) iff `x` ∈ `S` or `x` is a limit point of `S`.

The set `S` is closed if and only if Cl(`S`) = `S`.
In particular, the closure of the empty set is the empty set, and the closure of `X` itself is `X`.
The closure of an intersection of sets is always a subset of (but need not be equal to) the intersection of the closures of the sets.
In a union of finitely many sets, the closure of the union and the union of the closures are equal; for infinitely many sets, this need not be the case.
However in any case, the closure of a union of sets is always a superset of the union of the closures of the sets.
Since zero is a finite number and the union of zero sets is the empty set, this is another way to see that the empty set is its own closure; that is, the empty set is closed.

The closure of the set `S` is equal to the complement of the interior of the complement of `S`.

The subset `S` is dense in `X` iff Cl(`S`) = `X`.

If `A` is a subspace of `X` containing `S`, then the closure of `S` computed in `A` is equal to the intersection of `A` and the closure of `S` computed in `X`: Cl_{A}(`S`) = `A` ∩ Cl_{X}(`S`). In particular, `S` is dense in `A` iff `A` is a subset of Cl_{X}(`S`).