Completely-regular space

A topological space in which any two sets one of which is closed while the other consists of a single point can be functionally separated (cf. Separation axiom). Completely-regular spaces in which all one-point sets are closed (i.e. completely-regular $T_1$-spaces) are frequently called Tikhonov spaces. They form one of the most important classes of topological spaces, which is distinguished by several special properties and is very often encountered in the applications of topology to other branches of mathematics. For instance, the space of any topological group is a completely-regular space, but need not be a normal space. All Tikhonov spaces are Hausdorff spaces and may be defined as spaces with (Hausdorff) compactifications, i.e. as (even everywhere-dense) subspaces of compacta. Among the compactifications of a given space there is a unique (up to a homeomorphism) maximal or Stone–Čech compactification, which may be continuously mapped onto any (Hausdorff) compactification of the given space so that each one of the points of the given space is mapped into itself.

A direct definition of Tikhonov spaces, without recourse to real numbers and functions [3], is based on two conjugate bases of the space — an open base $\mathfrak{B}$ and a closed base $\mathfrak{A}$; that these bases are conjugate means that each base consists of sets complementary to the sets from the other base. Such a pair of conjugate bases $(\mathfrak{B},\mathfrak{A})$ is called regular if it satisfies the following conditions: 1) any two disjoint closed sets of $\mathfrak{A}$ have disjoint neighbourhoods belonging to $\mathfrak{B}$; and 2) $\mathfrak{A}$ is a network, i.e. for any point $x\in X$ and any neighbourhood $O_x\in\mathfrak{B}$ there exists an element $B\in\mathfrak{B}$ such that $X\setminus\{x\} \supset B \supset X\setminus O_x$. For a $T_1$-space to be completely regular it is necessary and sufficient for it to have at least one pair of conjugate bases (Zaitsev's theorem).

References

[1]	P.S. Aleksandrov, "Einführung in die Mengenlehre und die Theorie der reellen Funktionen" , Deutsch. Verlag Wissenschaft. (1956) (Translated from Russian)
[2]	L.S. Pontryagin, "Topological groups" , Princeton Univ. Press (1958) (Translated from Russian)
[3]	J.L. Kelley, "General topology" , Springer (1975)
[4]	P.S. Aleksandrov, B.A. Pasynkov, "Introduction to dimension theory" , Moscow (1973) (In Russian)
[5]	V.I. Zaitsev, "On the theory of Tikhonov spaces" Vestnik Moskov. Univ. Ser. I Math. Mekh. , 22 : 3 (1967) pp. 48–57 (In Russian) (English abstract)

Comments

Condition 2) above can also be formulated as: 2') $\mathfrak{A}$ is a network, i.e. for any point $x\in X$ and any neighbourhood $O_x\in \mathfrak{B}$ there is an element $A \in \mathfrak{A}$ such that $x \in A \subset O_x$.

Internal characterizations of complete regularity have been obtained by many authors. All are much like Zaitsev's result cited above. The one in [3] is identical to Zaitsev's; see also exercise 1.5.G in [a6].

References

[a1]	J. Kerstan, "Eine Charakterisierung der vollständig regulären Räume" Math. Nachr. , 17 (1958) pp. 27–46
[a2]	O. Frink, "Compactifications and semi-normal spaces" Amer. J. Math , 86 (1964) pp. 602–607
[a3]	E.F. Steiner, "Normal families and completely regular spaces" Duke Math. J. , 33 (1966) pp. 743–745
[a4]	J. de Groot, J.M. Aarts, "Complete regularity as a separation axiom" Canad. J. Math. , 21 (1969) pp. 96–105
[a5]	H. Brandenburg, A. Mysior, "A short proof of an internal characterization of complete regularity" Canad. Math. Bull. , 27 (1984) pp. 461–462
[a6]	R. Engelking, "General topology" , PWN (1977)