In mathematics, a closed point of a topological space is a point whose singleton is closed. In many areas of geometry and topology, all spaces under consideration are T<sub>1</sub> spaces that only have closed points. The distinction between closed and non-closed points is most often made in algebraic geometry, where schemes can have non-closed points.
If is a topological space, a point is called closed if the singleton is closed. An equivalent statement is that the closure only contains .
The closed points of a space can also be defined using the specialization preorder on . Given points , specializes to if . This means that the closed points of a topological space are those that specialize to no point except themselves, that is, the most specific points.
Spaces where every point is closed, called T<sub>1</sub> spaces, are common. In most branches of mathematics, it is rare to encounter spaces that have any non-closed points. Many mathematicians regard such spaces as somewhat strange. For example, if specializes to , the constant sequence converges to (as well as ).
In algebraic geometry, schemes usually have many non-closed points, including points whose closure is rather large. In particular, every irreducible component of a scheme is of the form for some . This can make the study of schemes easier since some properties of extend to the entirety of .
In any scheme that is locally of finite type over a field, the set of closed points is dense. In particular, this is true for schemes that correspond to algebraic varieties. This is not always the case, even for an affine scheme. For example, the spectrum of a discrete valuation ring is (topologically) the aforementioned Sierpià Âski space. Nonempty quasi-compact schemes (and in particular affine schemes) must have at least one closed point. However, there are schemes without any closed points at all, including irreducible schemes.
In any scheme that is locally of finite type over a field , the residue field is finite over at closed points and transcendental over at non-closed points. In particular, if is algebraically closed, the closed points are exactly those where the residue field is itself. This implies that every -rational point is closed, and if is algebraically closed then the closed points are exactly the -rational points. In a scheme of finite type over , the closed points are exactly the points where the residue field is finite, and each finite field is the residue field at only finitely many points. This makes it possible to define the arithmetic zeta function of such a scheme.
Let be affine scheme (or equivalently, a spectral space). is normal if and only if its closed points can be separated by neighborhoods. If the space of closed points of is connected, is connected too, and the converse holds if is normal. If is normal, the space of closed points of is compact (and Hausdorff). A normal affine scheme is simply the spectrum of a commutative Gelfand ring, so these are in fact properties of the maximal spectra of such rings.
A locally closed point, or a Goldman point, is a point such that the singleton is locally closed. This is equivalent to the condition that is isolated in . Every closed point is locally closed.
Unlike the case of closed points, the locally closed points are dense in every affine scheme.