Kolmogorov axiom

2020 Mathematics Subject Classification: Primary: 54D10 [MSN][ZBL]

$T_0$-axiom

The weakest of all separation axioms in general topology; introduced by A.N. Kolmogorov. A topological space satisfies this axiom, i.e. it is a $T_0$-space or a Kolmogorov space, if for any two distinct points of the space there exists an open set containing one of the points but not containing the other. If it is required that each of the two (arbitrarily given) points be contained in an open set not containing the other, then one obtains the next stronger separation axiom, called the $T_1$-axiom; topological spaces satisfying it are called $T_1$-spaces. The simplest example of a $T_0$-space that is not a $T_1$-space is the two-point space $\{a,b\}$ with open sets $\{ \emptyset, \{a\}, \{a,b\} \}$.

In a $T_0$-space, a singleton need not be closed; $T_1$-spaces can be defined as $T_0$-spaces for which all singletons are closed. A $T_0$-space in which the intersection of any number of open sets is open is called a discrete space (in the broad sense). It is precisely in these spaces that the closure of the union of any family of sets is the same as the union of the closures of these sets. In any discrete space and even in any Kolmogorov space one can define a (partial) order between its points $x$ and $y$: $x \sqsubseteq y$ if $x$ is contained in the closure of the singleton consisting of the point $y$. Conversely, if one defines in an arbitrary partially ordered set the closure of any point $x$ as the set of all points $x' \le x$ and takes for the closure of a set the union of the closures of all of its points, then one obtains a discrete space. Thus, the study of discrete spaces is equivalent to that of partially ordered sets. Simplicial (and more general) complexes in combinatorial topology are important examples of discrete spaces: For two simplices $x$, $y$ the order relation $x \le y$ means that $x$ is a face (possibly improper) of the simplex $y$.

References

[1]	P.S. Aleksandrov, "Einführung in die Mengenlehre und die allgemeine Topologie" , Deutsch. Verlag Wissenschaft. (1984) (Translated from Russian)

Comments

"Natural" spaces which are $T_0$ but, as a rule, not $T_1$ are the affine schemes $\mathrm{Spec}(A)$, for $A$ a commutative ring with identity; cf. Affine scheme.

"Discrete spaces in the broad sense" , mentioned above, are commonly called Aleksandrov-discrete spaces; they were first studied in [a1].

References

[a1]	P.S. [P.S. Aleksandrov] Alexandrov, "Diskrete Räume" Mat. Sb. , 1 (43) (1937) pp. 501–519

Comments

The partial order $y \in \overline{\{x\}}$ is the relation of specialization of a point in a topological space: this relation is a partial order if and only if the space is $T_0$.

References

[b1]	Steven Vickers Topology via Logic Cambridge Tracts in Theoretical Computer Science 5 Cambridge University Press (1989) ISBN 0-521-36062-5 Zbl 0668.54001