In topology, the closure of a subset S of points in a topological space consists of all points in S together with all limit points of S. The closure of S may equivalently be defined as the union of S and its boundary, and also as the intersection of all closed sets containing S. Intuitively, the closure can be thought of as all the points that are either in S or "very near" S. A point which is in the closure of S is a point of closure of S. The notion of closure is in many ways dual to the notion of interior.
[1] Note that this definition does not depend upon whether neighbourhoods are required to be open.
The definition of a point of closure of a set is closely related to the definition of a limit point of a set.
has more strict condition than a point of closure of
can be defined using any of the following equivalent definitions: The closure of a set has the following properties.
is the set of all limits of all convergent sequences of points in
For a general topological space, this statement remains true if one replaces "sequence" by "net" or "filter" (as described in the article on filters in topology).
Note that these properties are also satisfied if "closure", "superset", "intersection", "contains/containing", "smallest" and "closed" are replaced by "interior", "subset", "union", "contained in", "largest", and "open".
It is useful to distinguish between the interior and the surface of the sphere, so we distinguish between the open 3-ball (the interior of the sphere), and the closed 3-ball – the closure of the open 3-ball that is the open 3-ball plus the surface (the surface as the sphere itself).
These examples show that the closure of a set depends upon the topology of the underlying space.
is the set of rational numbers, with the usual relative topology induced by the Euclidean space
has no well defined closure due to boundary elements not being in
to be the set of real numbers and define the interval in the same way then the closure of that interval is well defined and would be the set of all real numbers greater than or equal to
then a topological space is obtained by defining the closed sets as being exactly those subsets
of these subsets form the open sets of the topology).
in the sense that and also Therefore, the abstract theory of closure operators and the Kuratowski closure axioms can be readily translated into the language of interior operators by replacing sets with their complements in
In general, the closure operator does not commute with intersections.
However, in a complete metric space the following result does hold: Theorem[7] (C. Ursescu) — Let
be a sequence of subsets of a complete metric space
from the definition of the subspace topology, there must exist some set
In words, this result shows that the closure in
can be computed "locally" in the sets of any open cover of
In this way, this result can be viewed as the analogue of the well-known fact that a subset
between topological spaces is continuous if and only if the preimage of every closed subset of the codomain is closed in the domain; explicitly, this means:
then this terminology allows for a plain English description of continuity:
Thus continuous functions are exactly those functions that preserve (in the forward direction) the "closeness" relationship between points and sets: a function is continuous if and only if whenever a point is close to a set then the image of that point is close to the image of that set.
One may define the closure operator in terms of universal arrows, as follows.
may be realized as a partial order category
in which the objects are subsets and the morphisms are inclusion maps
This category — also a partial order — then has initial object