1 Basic definition

A closed point is a point in a topological space whose singleton set is closed. In other words, the set containing only that point is a closed subset of the space. This is a simple but useful notion because it ties a point’s local behavior to the global separation properties of the surrounding space.

Closed points appear in many areas of mathematics. In ordinary topology, they often behave like points that can be distinguished from every other point by open neighborhoods. In algebraic geometry and related fields, the term is also used in a more specialized way, where closed points may correspond to maximal algebraic objects in a geometric space.

1.1 Closed sets and singletons

For a point x in a topological space X, the singleton {x} may or may not be closed. When it is closed, x is called a closed point. This means the complement X \ {x} is open, so the point can be removed while leaving an open remainder.

The condition is purely topological and does not require any metric or coordinate description. It depends only on the collection of open sets in the space.

1.2 Point as a closed subset

A point can be viewed as a subset consisting of one element. Under this viewpoint, the question is whether that one-element subset is closed. If it is, the point behaves like a closed subspace with no additional points attached to it through closure.

This perspective is especially helpful in comparing points with other types of subsets. It also connects the notion of a closed point with broader ideas about closed sets and closure operators.

1.3 Equivalent formulations

Several statements are equivalent to a point being closed. These formulations are often used interchangeably, depending on the context.

1.3.1 Complement is open

A point x is closed if and only if the complement of {x} is open. This is simply the definition of a closed set applied to a singleton. It is often the most direct way to verify the property.

1.3.2 Closure of the point

A point x is closed if and only if the closure of {x} is exactly {x}. In that case, no other point lies in the closure of x. Said differently, x has no nontrivial limit behavior inside the space.

2 Examples

Closed points can be found in many familiar settings, though their prevalence depends strongly on the space.

2.1 In discrete spaces

In a discrete space, every subset is open and closed. Therefore every point is closed. This is the simplest case, and it shows that the notion is automatic when the topology separates points as strongly as possible.

2.2 In familiar topological spaces

In a Hausdorff space, every point is closed. Common examples include the real line with its usual topology, Euclidean spaces, and manifolds. The reason is that points can be separated by neighborhoods, which forces singletons to be closed.

In spaces that are not Hausdorff, some points may fail to be closed. For instance, points can have closures containing several elements, so a singleton need not be closed.

2.3 In non-Hausdorff spaces

Non-Hausdorff spaces often provide examples where closed points and non-closed points coexist. In such spaces, a point may lie in the closure of another point, reflecting a lack of separation. This makes closed points useful for understanding how far the space is from being Hausdorff.

A point that is not closed may still be important structurally, because its closure can encode specialization relations among points.

2.4 In algebraic geometry

In algebraic geometry, closed points on an affine or projective variety often correspond to maximal ideals or to geometric points with no proper generalizations. Over an algebraically closed field, many classical varieties have closed points that match the intuitive notion of ordinary points.

The precise meaning depends on the space being studied. On schemes, the classification of points is more subtle, and a closed point need not be the same as a point that is closed in the ambient topological sense alone.

3 Characterizations

Closed points admit several useful characterizations that connect topology with order theory and closure theory.

3.1 In terms of separation axioms

In a T1 space, every point is closed. This means the T1 axiom is exactly the statement that all singletons are closed. More generally, spaces with stronger separation properties automatically have many or all closed points.

When the space is not T1, closed points may still exist individually. Their presence indicates that at least some points are separated from the rest of the space by open sets.

3.2 In terms of closure operators

Using the closure operator cl, a point x is closed precisely when cl({x}) = {x}. This says that the point is already equal to the smallest closed set containing it. No additional points are forced into the closure.

This description is often convenient in proofs involving convergence or accumulation. It translates the property into a statement about fixed points of the closure operation.

3.3 In terms of specialization order

In a topological space, one can define a specialization preorder by x ≤ y if x lies in the closure of {y}. A point is closed when it is maximal in this order, meaning no strictly larger specialization lies above it.

This viewpoint is especially important in algebraic geometry and finite topological spaces. It shows that closed points are those with no proper generizations in the specialization structure.

4 Properties

Closed points satisfy several basic stability and comparison properties.

4.1 Isolated points versus closed points

An isolated point is one whose singleton is open. Every isolated point is closed, because if {x} is open then its complement is closed, and in many standard spaces this also implies the singleton is closed as a subset. However, being closed does not imply being isolated. A point may be closed without being open, as in the real line.

Thus isolatedness is stronger than closedness. The former concerns openness of the singleton, while the latter concerns its complement.

4.2 Closed points in Hausdorff spaces

In any Hausdorff space, all points are closed. This is one of the most familiar consequences of the separation axiom. It is a major reason why closed points are not usually emphasized in elementary Euclidean settings, where the property holds automatically.

The converse is not true: a space may have all points closed and still fail to be Hausdorff. The distinction lies in whether distinct points can be separated by disjoint neighborhoods.

4.3 Stability under subspaces

If x is a closed point in a subspace Y, then {x} is closed in Y. This need not imply that {x} is closed in the larger space X, since closedness can change when passing between a subspace and its ambient space.

Conversely, if x is closed in X, then it remains closed in any subspace containing x. This follows because the intersection of a closed set with a subspace is closed in the subspace topology.

4.4 Behavior under continuous maps

Continuous maps do not generally preserve closed points in either direction. The image of a closed point need not be closed, and a preimage point over a closed point may fail to be closed. Additional hypotheses, such as special separation properties or finite-type conditions, are often needed to obtain stronger behavior.

Nevertheless, closed points interact well with closed maps and with maps that reflect separation. In such cases, closedness can be transported more reliably between spaces.

5 Closed points in algebraic geometry

In algebraic geometry, closed points play a central role in translating geometric intuition into algebraic data.

5.1 Relation to maximal ideals

For an affine scheme Spec A, closed points correspond to maximal ideals of the ring A. This is one of the foundational dictionary entries of scheme theory. The point associated with a maximal ideal has no further specialization inside Spec A.

This relationship makes closed points a bridge between geometry and commutative algebra. Algebraic questions about ideals become geometric questions about points.

5.2 Closed points on varieties

On an affine or projective variety over an algebraically closed field, closed points often correspond to the classical geometric points of the variety. In this setting, they match the intuitive notion of ordinary solutions to polynomial equations.

Over non-algebraically closed fields, the picture is more subtle. A closed point may correspond to an orbit of geometric points under field extension, and its residue field may be a finite extension of the base field.

5.3 Closed points on schemes

Schemes can have points that are not closed, such as generic points of irreducible components. Closed points are the points at the “ends” of the specialization relation, while non-closed points may encode more generic algebraic information.

5.3.1 Residue fields

Each point of a scheme has an associated residue field. For a closed point, this field is often finite over the base field in common geometric situations, though not always. The residue field records the algebraic content of the point.

5.3.2 Generic points and specialization

A generic point lies in the closure of many other points and is typically not closed. Closed points are maximal with respect to specialization, while generic points are minimal in the opposite direction. Together they describe the closure structure of the scheme.

Several nearby notions help clarify what closed points are and are not.

6.1 Open points

An open point is a point whose singleton set is open. Such points are isolated. Every open point is closed in many standard contexts, but openness is a stronger condition than closedness.

6.2 Limit points

A limit point is a point that can be approached by other points of a set. A closed point, by contrast, has no extra points in the closure of its singleton. These notions are therefore complementary in spirit, though they apply in different ways.

6.3 Generic points

A generic point represents an irreducible closed set and is often highly non-closed. It stands at the opposite end of the specialization order from a closed point. Generic points are especially important in scheme theory and in irreducible topological spaces.

6.4 Closed subsets

A closed point is a special case of a closed subset, namely one consisting of a single element. Studying closed subsets provides the broader framework in which closed points are defined. Many properties of closed points are immediate consequences of the general theory of closed sets.

7 Applications

Closed points are used in topology, algebraic geometry, and order-theoretic settings to describe structure at the level of individual points.

7.1 Topology

In topology, closed points help identify how sharply a space separates its elements. They are useful in classifying spaces by separation axioms, understanding convergence, and studying quotient spaces or non-Hausdorff examples.

7.2 Algebraic geometry

In algebraic geometry, closed points are essential for describing geometric solutions, residue fields, and the link between points and maximal ideals. They provide a concrete endpoint for specialization and are often the points of primary interest in geometric applications.

7.3 Order-theoretic interpretations

Via specialization order, closed points become maximal elements in a preorder. This interpretation is valuable in finite topological spaces, spectral spaces, and other settings where topology and order interact closely. It offers a compact way to describe the end points of closure relations.