A. A. KUBENSKY

ON FUNCTIONALLY CLOSED SPACES

(Presented by Academician P. S. Aleksandrov on 20 VI 1957)

A completely regular space (P) is called functionally closed if, for every proper extension (\widetilde P) of the space (P), there exists a continuous real-valued function defined on (P) that cannot be extended* to (\widetilde P). Such spaces, under the name of (Q)-spaces, were first considered by Hewitt ((^1)).

We shall call an extension (\widetilde P) of the space** (P) regular if every continuous function defined on (P) can be continuously extended to the extension (\widetilde P). We shall call the zero-set of a function (f), defined on the space (P), the complete preimage (f^{-1}(0)) of the number (0), and the zero-set of the space (P) a set that is the zero-set of some function.

Lemma 1. Let (\widetilde P) be a regular extension of the space (P); for any continuous function (f) defined on the space (P), its extension has as its zero-set the closure in (\widetilde P) of the zero-set of the function (f).

Theorem 1. In order that an extension (\widetilde P) of the space (P) be regular, it is necessary that the closure of the intersection of any countable sequence of zero-sets of the space (P) coincide with the intersection of their closures in (\widetilde P), and it is sufficient that, for any countable sequence of zero-sets with empty intersection, the sequence of closures of these sets in (\widetilde P) also have empty intersection.

Proof. Necessity. Let ({f_i}) be a countable set of continuous functions defined on (P), and suppose that everywhere (|f_i|\leq 1/2^i). Then the function

[
\varphi=\sum_{i=1}^{\infty} f_i
]

is continuous, and the intersection of the zero-sets (N(f_i)) of the functions (f_i) is its zero-set. From Lemma 1 we obtain

[
\widetilde P\left[\bigcap_i N(f_i)\right]=\bigcap_i \widetilde P[N(f_i)],
]

as required.

Sufficiency. Let (I) be the number line. To each point (x) of (\widetilde P\setminus P) we assign the family (\mathfrak B(x)) of all such zero-sets (F) of the space (P) that (x\in \widetilde P[F]). For any continuous function (f), defined on (P), and for any point (x) of (\widetilde P\setminus P), set

[
\widetilde f(x)=\bigcap_{F\in \mathfrak B(x)} I[f(F)].
]

It turns out that the extension of the function (f) thus defined is continuous on (P). The theorem is proved.

* One says that a function (f), defined on the space (P), is continuously extendable to a space (S\supset P) if there exists a continuous function (\widetilde f), defined on (S), identically equal to (f) on (P). The function (\widetilde f) is called an extension of the function (f).

** By a space we shall always mean a completely regular space.

We shall call a nonempty maximal centered family of zero-sets of a space (P) perfect if the intersection of any of its countable subfamilies is nonempty. An extension (\widetilde P) of a space (P) will be called a (\vartheta)-extension if: 1) (\widetilde P) is a regular extension of the space (P); 2) (\widetilde P) is functionally closed.

For every space (P), denote by (\Omega(P)) the family of all perfect families of this space that have empty intersection. For every zero-set (F) of the space (P), denote by (\tau(F)) the set consisting of all points of this set (F) and all perfect families from (\Omega(P)) that contain the set (F) as an element.

Theorem 2. The set (\vartheta P), consisting of all points of the space (P) and of all elements of the family (\Omega(P)), is a (\vartheta)-extension of the space (P), if one defines in it a topology by taking as closed sets all possible intersections of sets of the form (\tau(F)).

Proof. The fact that the extension (\vartheta P) of the space (P) so obtained is regular follows directly from Theorem 1. The functional closedness of the extension (\vartheta P) follows directly from the sufficiency of the following (Hewitt’s ((^1))) condition for functional closedness:

Theorem 3. In order that a space (P) be functionally closed, it is necessary and sufficient that every perfect family of the space (P) have nonempty intersection.

Proof. The sufficiency of the stated condition again follows from Theorem 1, if one first observes that for every point (x) of (\widetilde P \setminus P), where (\widetilde P) is an arbitrary regular extension of the space (P), the family (\mathfrak B(x)) is a nonempty maximal centered family of zero-sets of the space (P). The necessity of the condition follows directly from Theorem 2.

From Theorem 3 there follows directly:

Theorem 4. For every space (P) there exists a unique (up to a topological mapping identical on (P)) (\vartheta)-extension.

We next clarify the connection between the notion of functional closedness and topological completeness in the sense of Dieudonné ((^2)). The latter is based on the notion of a uniform space and a uniform structure in the sense of A. Weil ((^3)). For our purposes it is more convenient to use the definition of a uniform structure as a certain system of coverings, equivalent to the original one ((^4)), since we shall need the following completeness criterion for a uniform space, found by Yu. M. Smirnov:

A system (\xi) of subsets of the space (P_\Sigma) with covering structure (\Sigma) is called a (\Sigma)-system if, for every covering (\gamma) of the structure (\Sigma), there is in the system (\xi) a set (A) contained in some element of the covering (\gamma). The space (P) is complete relative to the structure (\Sigma) (i.e. the uniform space (P_\Sigma) is complete) if and only if every centered closed (\Sigma)-system has nonempty intersection (((^5)), p. 431).

Consider the following systems of coverings of the space (P): (\sigma_0)—the system of countable open normal* locally finite coverings; (\sigma'_0)—the system of countable open normal coverings; (\sigma_1)—the system of open normal locally finite coverings. Each of the systems considered generates a system (\Sigma_0), respectively (\Sigma'_0) and (\Sigma_1), consisting by definition of all those coverings of the space (P) in each—

* A covering (\gamma), consisting of sets (\Gamma_\lambda) of the space (P), is called normal if for each (\Gamma_\lambda) one can choose a set (A_\lambda \subseteq \Gamma_\lambda), functionally separated from (P \setminus \Gamma_\lambda), such that (\bigcup_\lambda A_\lambda = P).

each of which contains some cover of the system under consideration.

Lemma 2. The systems (\Sigma_0), (\Sigma'_0), and (\Sigma_1) are uniform structures of the space (P), and (\Sigma_0=\Sigma'_0).

Theorem 5. Every functionally closed space (P) is complete with respect to the structure (\Sigma_1).

The proof is based on the fact that, if the uniform space (P_{\Sigma_1}) is not complete, then the space of its completion turns out to be a proper extension of the space (P), distinct from (P) itself.

Theorem 6. Every space complete with respect to the structure (\Sigma_0) is functionally closed.

Proof. Let (\xi) be a perfect family of the space (P), complete with respect to the structure (\Sigma_0). Since (\xi) turns out to be a centered closed (\Sigma)-system, its intersection is nonempty by the stated completeness criterion of Yu. M. Smirnov. By Theorem 3 the space (P) is functionally closed, as was required to prove.

Theorem 7. In order that a space be functionally closed, it is necessary and sufficient that it be complete with respect to the structure (\Sigma_0).

We shall call a cardinal number (\mathfrak m) (F)-regular if the discrete space of cardinality (\mathfrak m) is functionally closed.

Theorem 8. Every normal space of (F)-regular cardinality, complete with respect to the structure (\Sigma_1), is functionally closed*.

For normal spaces the structure (\Sigma_1) is the maximal** structure.

Hence it follows:

Theorem 9. In order that a normal space of attainable cardinality*** be functionally closed, it is necessary and sufficient that it be topologically complete (i.e. complete with respect to the maximal uniform structure (\Sigma_1)).

CITED LITERATURE

({}^{1}) E. Hewitt, Trans. Am. Math. Soc., 64, 45 (1948).
({}^{2}) J. Dieudonné, Ann. Sci. de l’école norm. sup., 56, 4, 277 (1939).
({}^{3}) A. Weil, Actualités scientifiques et industrielles, 551, Sur les espaces à structure uniforme et sur la topologie générale, 1938.
({}^{4}) Yu. M. Smirnov, Mat. sborn., 31, 563 (1952).
({}^{5}) Yu. M. Smirnov, Tr. Mosk. matem. obshch., 4, 421 (1955).
({}^{6}) M. Katetov, Fund. Math., 38, 73 (1951).

* Cf. Katetov’s theorem ((({}^{6})), p. 82).
* A uniform structure of covers is maximal if it contains every other uniform structure of the space (P) as a substructure.
** A cardinal number is attainable if it is less than the first unattainable cardinal number. A cardinal number (n>\aleph) is unattainable if it cannot be represented as a sum of powers (2^{\mathfrak k_\lambda}), the number of which is less than (n).