Kolmogorov space

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topological spaces
Kolmogorov classification
T0	(Kolmogorov)
T1	(Fréchet)
T2	(Hausdorff)
T2½	(Urysohn)
completely T2	(completely Hausdorff)
T3	(regular Hausdorff)
T3½	(Tychonoff)
T4	(normal Hausdorff)
T5	(completely normal  Hausdorff)
T6	(perfectly normal  Hausdorff)
History

In

.

This condition, called the T₀ condition, is the weakest of the separation axioms. Nearly all topological spaces normally studied in mathematics are T₀ spaces. In particular, all T₁ spaces, i.e., all spaces in which for every pair of distinct points, each has a neighborhood not containing the other, are T₀ spaces. This includes all T₂ (or Hausdorff) spaces, i.e., all topological spaces in which distinct points have disjoint neighbourhoods. In another direction, every sober space (which may not be T₁) is T₀; this includes the underlying topological space of any scheme. Given any topological space one can construct a T₀ space by identifying topologically indistinguishable points.

T₀ spaces that are not T₁ spaces are exactly those spaces for which the

.

Definition

A T₀ space is a topological space in which every pair of distinct points is

that contains one of these points and not the other. More precisely the topological space X is Kolmogorov or ${\displaystyle \mathbf {T} _{0}}$ if and only if:[1]

If ${\displaystyle a,b\in X}$ and ${\displaystyle a\neq b}$, there exists an open set O such that either ${\displaystyle (a\in O)\wedge (b\notin O)}$ or ${\displaystyle (a\notin O)\wedge (b\in O)}$.

Note that topologically distinguishable points are automatically distinct. On the other hand, if the

then the points x and y must be topologically distinguishable. That is,

separated ⇒ topologically distinguishable ⇒ distinct

The property of being topologically distinguishable is, in general, stronger than being distinct but weaker than being separated. In a T₀ space, the second arrow above also reverses; points are distinct if and only if they are distinguishable. This is how the T₀ axiom fits in with the rest of the separation axioms.

Examples and counter examples

Nearly all topological spaces normally studied in mathematics are T₀. In particular, all Hausdorff (T₂) spaces, T₁ spaces and sober spaces are T₀.

Spaces which are not T₀

• A set with more than one element, with the trivial topology. No points are distinguishable.
• The set R² where the open sets are the Cartesian product of an open set in R and R itself, i.e., the product topology of R with the usual topology and R with the trivial topology; points (a,b) and (a,c) are not distinguishable.
• The space of all
  Lebesgue integral
  ${\displaystyle \left(\int _{\mathbb {R} }|f(x)|^{2}\,dx\right)^{\frac {1}{2}}<\infty }$. Two functions which are equal almost everywhere are indistinguishable. See also below.

Spaces which are T₀ but not T₁

• The
  prime spectrum of a commutative ring R, is always T₀ but generally not T₁. The non-closed points correspond to prime ideals which are not maximal. They are important to the understanding of schemes
  .
• The particular point topology on any set with at least two elements is T₀ but not T₁ since the particular point is not closed (its closure is the whole space). An important special case is the Sierpiński space which is the particular point topology on the set {0,1}.
• The excluded point topology on any set with at least two elements is T₀ but not T₁. The only closed point is the excluded point.
• The Alexandrov topology on a partially ordered set is T₀ but will not be T₁ unless the order is discrete (agrees with equality). Every finite T₀ space is of this type. This also includes the particular point and excluded point topologies as special cases.
• The
  totally ordered set
  is a related example.
• The overlapping interval topology is similar to the particular point topology since every non-empty open set includes 0.
• Quite generally, a topological space X will be T₀ if and only if the
  partial order
  . However, X will be T₁ if and only if the order is discrete (i.e. agrees with equality). So a space will be T₀ but not T₁ if and only if the specialization preorder on X is a non-discrete partial order.

Operating with T₀ spaces

Commonly studied topological spaces are all T₀. Indeed, when mathematicians in many fields, notably

. This space should become a

zero

. The standard solution is to define L²(R) to be a set of equivalence classes of functions instead of a set of functions directly. This constructs a quotient space of the original seminormed vector space, and this quotient is a normed vector space. It inherits several convenient properties from the seminormed space; see below.

In general, when dealing with a fixed topology T on a set X, it is helpful if that topology is T₀. On the other hand, when X is fixed but T is allowed to vary within certain boundaries, to force T to be T₀ may be inconvenient, since non-T₀ topologies are often important special cases. Thus, it can be important to understand both T₀ and non-T₀ versions of the various conditions that can be placed on a topological space.

The Kolmogorov quotient

Topological indistinguishability of points is an

. Categorically, Kolmogorov spaces are a reflective subcategory of topological spaces, and the Kolmogorov quotient is the reflector.

Topological spaces X and Y are Kolmogorov equivalent when their Kolmogorov quotients are homeomorphic. Many properties of topological spaces are preserved by this equivalence; that is, if X and Y are Kolmogorov equivalent, then X has such a property if and only if Y does. On the other hand, most of the other properties of topological spaces imply T₀-ness; that is, if X has such a property, then X must be T₀. Only a few properties, such as being an

, are exceptions to this rule of thumb. Even better, many defined on topological spaces can be transferred between X and KQ(X). The result is that, if you have a non-T₀ topological space with a certain structure or property, then you can usually form a T₀ space with the same structures and properties by taking the Kolmogorov quotient.

The example of L²(R) displays these features. From the point of view of topology, the seminormed vector space that we started with has a lot of extra structure; for example, it is a

that are compatible with the topology. Also, there are several properties of these structures; for example, the seminorm satisfies the are indistinguishable with this topology. When we form the Kolmogorov quotient, the actual L²(R), these structures and properties are preserved. Thus, L²(R) is also a complete seminormed vector space satisfying the parallelogram identity. But we actually get a bit more, since the space is now T₀. A seminorm is a norm if and only if the underlying topology is T₀, so L²(R) is actually a complete normed vector space satisfying the parallelogram identity—otherwise known as a Hilbert space. And it is a Hilbert space that mathematicians (and of square integrable functions that differ on sets of measure zero, rather than simply the vector space of square integrable functions that the notation suggests.

Removing T₀

Although norms were historically defined first, people came up with the definition of seminorm as well, which is a sort of non-T₀ version of a norm. In general, it is possible to define non-T₀ versions of both properties and structures of topological spaces. First, consider a property of topological spaces, such as being

. (Again, there is a more direct definition of pseudometric.)

In this way, there is a natural way to remove T₀-ness from the requirements for a property or structure. It is generally easier to study spaces that are T₀, but it may also be easier to allow structures that aren't T₀ to get a fuller picture. The T₀ requirement can be added or removed arbitrarily using the concept of Kolmogorov quotient.

See also

• Sober space

References

• Lynn Arthur Steen and J. Arthur Seebach, Jr., Counterexamples in Topology. Springer-Verlag, New York, 1978. Reprinted by Dover Publications, New York, 1995.
  ISBN 0-486-68735-X
  (Dover edition).