Closed point

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 T₁ 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 ${\displaystyle X}$ is a topological space, a point ${\displaystyle x\in X}$ is called closed if the singleton ${\displaystyle \{x\}}$ is closed.^[1] The closed points of a topological space are the most specific points under the specialization preorder.^[2][3] This means that a point is non-closed if it specializes to at least one other point.

Closed points can be thought of as having specific positions.^[4] If ${\displaystyle x\in X}$ is non-closed, it is generic in the set ${\displaystyle {\overline {\{x\}}}}$ of points to which it specializes, which includes at least one point in addition to ${\displaystyle x}$. ${\displaystyle x}$ can then be pictured as a nonspecific point^[4] that is somewhere within ${\displaystyle {\overline {\{x\}}}}$ but has no particular position within it,^[5] or as a diffuse large point^[6] that contains all other points of ${\displaystyle {\overline {\{x\}}}}$.^[7] Both of these visualizations portray the fact that ${\displaystyle x}$ is "almost everywhere" in ${\displaystyle {\overline {\{x\}}}}$ at the same time, and it is near each other point of ${\displaystyle {\overline {\{x\}}}}$.^[5]

• A topological space is T₁ if and only if all of its points are closed.^[8] Many types of spaces are T₁,^[8] such as manifolds (including non-Hausdorff manifolds^[9]) and spaces with the cofinite topology.
• When an algebraic variety is considered as a scheme, every Zariski-closed subvariety of it is endowed with an additional point, which is generic in that subvariety. The original points of the variety are the closed points of the resulting scheme.^[10]
• In the spectrum of a commutative ring, the points are the prime ideals of the ring, and the closed points are the maximal ideals.^[1]
• In particular, the points of the spectrum of a principal ideal domain are the ideals generated by prime elements (defined up to a unit) and the zero ideal. The points that correspond to prime elements are closed, and the point that corresponds to zero is generic.^[11]
• The Sierpiński space has two points. One of them is closed, and the other is non-closed since it specializes to the first point.^[12]
• If two points are topologically indistinguishable, they specialize to each other and hence neither of them is closed.

In any scheme that is locally of finite type over a field, the set of closed points is dense.^[13] 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 Sierpiński space.^[14] Nonempty quasi-compact schemes (and in particular affine schemes) must have at least one closed point.^[15] However, there are schemes without any closed points at all,^[16] including irreducible schemes.^[15]

Let ${\displaystyle X}$ be affine scheme (or equivalently, a spectral space). ${\displaystyle X}$ is normal if and only if its closed points can be separated by neighborhoods.^[17] If the space of closed points of ${\displaystyle X}$ is connected, ${\displaystyle X}$ is connected too, and the converse holds if ${\displaystyle X}$ is normal.^[18] If ${\displaystyle X}$ is normal, the space of closed points of ${\displaystyle X}$ is compact (and Hausdorff).^[19] A normal affine scheme is simply the spectrum of a commutative Gelfand ring,^[20] so these are in fact properties of the maximal spectra of such rings.

A locally closed point, or a Goldman point, is a point ${\displaystyle x\in X}$ such that the singleton ${\displaystyle \{x\}}$ is locally closed. This is equivalent to the condition that ${\displaystyle x}$ is isolated in ${\displaystyle {\overline {\{x\}}}}$, and it is weaker than the condition of being closed.^[21]

Unlike the case of closed points, the locally closed points are dense in every affine scheme.^[22]