UDC 513.88:513.83+519.50+519.54

MATHEMATICS

A. I. VEKSLER

(P)-SETS IN TOPOLOGICAL SPACES

(Presented by Academician P. S. Aleksandrov on 16 I 1970)

1. We shall, as far as possible, adhere to the terminology of the monographs ((^{3,4})) and the survey article ((^{1})).

The concept of a (P)-point belongs to L. Gillman and M. Henriksen ((^{5})). A (P)-point is a point of a completely regular space (T) such that, in the intersection of any countable number of (open) neighborhoods of it, there is still one neighborhood.

It is well known, for example, that an arbitrary (t \in T) is a (P)-point if and only if every (x \in C(T)) is constant in some neighborhood of (t). If (t) is a (P)-point in (T) and (t \in A \subset T), then (t) is a (P)-point also in (A). A point (t \in T) is a (P)-point if and only if there exists only one prime ideal in the ring (C(T)) attached to (t), namely the ideal (M_t) of all functions vanishing at (t).

A natural generalization of the concept of a (P)-point is the concept of a (P)-set. A closed set (F \subset T) is called a (P)-set if, in the intersection of any countable number of open sets containing (F), there is still one such set.

For simplicity we shall consider (P)-sets only in bicompacts and assume that all spaces considered below (but not their subspaces) are bicompacts.

For an arbitrary bicompact (K), its (P)-sets can be characterized as follows: (F \subset K) is a (P)-set if and only if it is closed and every (x \in C(K)) that is constant on (F) is constant also on some neighborhood of (F). A closed (\Phi) is a (P)-set if and only if
[
I_{\Phi}={x \in C(K): x(\Phi)\equiv 0}
]
is the unique ideal of (C(K)) for which ({M_k: k \in \Phi}) is the collection of all maximal ideals containing this ideal.

Although (P)-sets had appeared episodically earlier in some works, the value of the very concept of a (P)-set was first, in our opinion, shown in the work of Z. T. Dikanova ((^{6})). Z. T. Dikanova considered (P)-sets in extremally disconnected bicompacts* and showed that (P)-sets arise naturally in the theory of semiordered spaces when solving certain problems concerning extended (K)-spaces. Recall that extended (K)-spaces are precisely those (K)-spaces that are isomorphic to spaces (C_{\infty}(Q)) of all continuous real functions on an extremally disconnected bicompact, which may take the values (\pm\infty) on nowhere dense subsets of (Q).

Apparently, it may now be regarded as a timely problem to study (P)-sets independently. Here, however, this problem is not considered. In this note, first, it is shown that the consideration of (P)-sets in quasiextremal bicompacts is natural in solving certain other problems of the theory of semiordered spaces, essentially different from

* Z. T. Dikanova additionally required that a (P)-set be nowhere dense. Under such a definition, however, an isolated point is not a (P)-set, although it is a (P)-point.

problems solved in (6) and relating to extended (K_\sigma)-spaces. Recall that a bicompactum (S) is called quasiextremal (or basically disconnected) if in it the closure of every open set of type (F_\sigma) is open; of course, an extremally disconnected bicompactum is quasiextremal. Every extended (K_\sigma)-space is isomorphic to the space (C_\infty(S)), and conversely, every function space of this type is an extended (K_\sigma)-space. Secondly, with the aid of certain results relating to extended (K_\sigma)-spaces and obtained earlier by the author, some properties of (P)-sets of an arbitrary quasiextremal bicompactum will be derived very simply.

1. A positive linear mapping (\varphi) of one (K)-linear (vector-lattice) structure (X) onto another (Y) is called a structural homomorphism if
   [
   \varphi(x_1 \vee x_2)=\varphi(x_1)\vee \varphi(x_2)
   \quad \text{and} \quad
   \varphi(x_1 \wedge x_2)=\varphi(x_1)\wedge \varphi(x_2).
   ]
   If (I) is an (l)-ideal which is the kernel of the homomorphism (\varphi), then the (K)-lineals (Y) and (X/I) are naturally isomorphic. A (K)-lineal (Z) is called Archimedean if from the inequalities (0\le nz\le z') ((n\in N)) it follows that (z=0); of course, a (K_\sigma)-space is Archimedean. In (2) the following was proved.

Theorem 0. Let (X) be an extended (K_\sigma)-space, and (I) its (l)-ideal. In order that the (K)-lineal (X/I) be Archimedean, it is necessary and sufficient that (I) be (\sigma)-closed, i.e., from ({x_n}\subset I) ((n\in N)) and the existence of
[
x=\sup x{x_n}
]
it follows that (x\in I). When this condition is fulfilled, (X/I) is an extended (K_\sigma)-space.

Let us now translate the first part of this theorem into topological language. In what follows, by the support of (x\in C_\infty(S)) on a quasiextremal (S) we shall mean the open-closed set
[
S_x={s\in S:\ x(s)\ne 0}.
]

Theorem 1. Let (S) be a quasiextremal bicompactum, (X=C_\infty(S)), and let (I) be an (l)-ideal in (X). In order that the (K)-lineal (X/I) be Archimedean, it is necessary and sufficient that
[
I=I_F={x\in X:\ x(F)\equiv 0}
]
for some (P)-set (F).

Proof. It is necessary to verify that the condition of the theorem is equivalent to the (\sigma)-closedness of the (l)-ideal (I).

Let (I) be a (\sigma)-closed (l)-ideal. As the required (F) take the intersection of the zero-sets of all functions in (I). Let (e\in X) be the unit (i.e. the function taking only the values 0 and 1) and let (e(F)\equiv 0). We show that (e\in I). By assumption, for any (s\in S_e\subset S\setminus F) there exists (x_s\in I) such that (x_s(s)\ne 0). Since (I) is an (l)-ideal, without loss of generality we may suppose that (x_s>0) and (x_s(s)>1). From the cover of the closed set (S_e) by the open sets
[
S_s={s'\in S:\ x_s(s')>1}
]
choose a finite subcover (S_{s_1},\ldots,S_{s_n}). Obviously,
[
0\le e\le \sup{x_{s_i}}\in I,
]
whence (e\in I).

Let us check that (I=I_F). It is enough to make sure that if (x\in X^+) and (z(F)\equiv 0), then (z\in I). It is known that any (z\in X^+) can be represented as the supremum of a sequence of positive linear combinations (z_n) of unit elements from (X). But, obviously, in the present case the supports of these unit elements do not intersect (F), i.e. these unit elements (and then also (z_n)) by what has already been proved belong to (I). And then
[
z=\sup{z_n}\in I
]
by virtue of the (\sigma)-closedness of (I). Let us verify that (F) is a (P)-set.

Let ({G_n}) be a sequence of open neighborhoods of the set (F). We check that there exists an open neighborhood contained in each (G_n). By virtue of the complete disconnectedness of (S), it suffices to restrict ourselves to the case of open-closed (G_n). Let (e_n) be the characteristic function of the set (S\setminus G_n). By what has already been proved, (e_n\in I). Let further (e=\sup{e_n}) and
[
G=S\setminus S_e.
]
Then (G) is open-closed,
[
G=\operatorname{Int}\bigl(\bigcap G_n\bigr),
]
and (e_n\in I) by virtue of the (\sigma)-closedness of (I). Hence (F\subset G). Thus (F) is a (P)-set.

We prove also conversely that every (l)-ideal (I=I_F), where (F) is a (P)-set, is (\sigma)-closed. If (z\in I^+), then
[
S_z=\bigcup{S_m:\ m\in N},
]
where
[
S_m={s\in S:\ z(s)>m^{-1}}.
]
But since the (S_m) are closed and (S_m\cap F=\Lambda), then also

$S_z \cap F = \Lambda$ (in view of the fact that $F$ is a $P$-set). Now let ${z_n} \subset I^+$ and $x = \sup {z_n}$. Then $S_{z_n} \cap F = \Lambda$ and $S_z = \bigcup S_{z_n}$. Again, since $F$ is a $P$-set, from the last two equalities it follows that $S_z \cap F = \Lambda$, i.e. $z(F) = 0$, $z \in I$. The theorem is proved.

We see that the problem of finding structurally homomorphic Archimedean images of the extended $K_\sigma$-space $X$ reduces to the problem of finding all $\sigma$-closed $l$-ideals in $X$; the second problem turns out to be equivalent to the problem of finding all $\sigma$-closed $l$-ideals in $C_\infty(S)$, and the latter problem turns out to be equivalent to the problem of finding all $P$-sets in the quasiextremal bicompactum $S$. Thus a problem from the theory of partially ordered spaces turns out to be equivalent to the topological problem of finding all $P$-sets of a quasiextremal bicompactum.

1. With the aid of Theorems 0 and 1 we shall establish a number of properties of $P$-sets in a quasiextremal bicompactum $S$. Some of them were proved by Z. T. Dikanova by a direct method for the case of an extremally disconnected $S$ in (⁶), or in her report at the seminar at Leningrad University. We shall call open-and-closed sets trivial $P$-sets, and the remaining $P$-sets nontrivial.

Theorem 2. For a quasiextremal bicompactum the following assertions hold:

a) Nontrivial $P$-sets in $S$ exist if and only if $S$ is of uncountable type (in other terminology, when its cellularity number is greater than $\aleph_0$).

b) If ${E_\xi}$ $(\xi \in \Xi)$ is a system of $P$-sets from $S$, then also $F = \overline{\bigcup F_\xi}$ is a $P$-set.

c) Every closed $\Phi$ contains the greatest $P$-set from $S$.

d) The family of all $P$-sets from $S$, partially ordered by inclusion, is a complete lattice with least element $\Lambda$ and greatest element $S$.

e) A closed $\Phi$ is a $P$-set if and only if the trace of every closed nowhere dense (c.n.d.) set of type $G_\delta$ from $S$ is n.n.d. also in $\Phi$ (and, obviously, is a closed set of type $G_\delta$ in $\Phi$).

f) A closed $\Phi$ is a $P$-set if and only if, for no open-and-closed $S_0 \subset S$ for which $S_0 \cap \Phi \ne \Lambda$, does this intersection immerse itself in a c.n.d. set from $S$ of type $G_\delta$.

Proof. a) In Theorem 1, by means of the equality $I = I_F$, a one-to-one correspondence was established between $\sigma$-closed $l$-ideals in the $K_\sigma$-space $X = C_\infty(S)$ and $P$-sets in $S$. If $F$ is open-and-closed, then this precisely means that $I_F$ is a component in $X$, i.e. a closed $l$-ideal. Thus the existence of nontrivial $P$-sets in $S$ is equivalent to the existence of $\sigma$-closed $l$-ideals in $X$ that are not components. But if $S$ is of countable type, then $X$ is also of countable type, and then every $\sigma$-closed $l$-ideal is a component in $X$. Hence in this case there are no nontrivial $P$-sets in $S$. Conversely, if $S$ has uncountable type, then let ${S_\xi}$ $(\xi \in \Xi)$ be an uncountable system of nonempty pairwise disjoint open-and-closed sets from $S$. Put $I_\xi = I_{S_\xi}$, and let $I$ be the $l$-ideal whose elements are all suprema in $X$ of finite or countable sets from $I_\xi$ $(\xi \in \Xi)$. From the $\sigma$-closedness of all $I_\xi$ follows the $\sigma$-closedness of $I$. But, obviously, $I$ is not a component. Therefore, if $I = I_F$, then $F$ will be a nontrivial $P$-set in $S$.

b) We have $I_F = \bigcap I_{F_\xi}$. But the intersection of any set of $\sigma$-closed $l$-ideals is itself a $\sigma$-closed $l$-ideal. Hence $F$ is a $P$-set.

c) Follows from b).

d) Follows from b) and c).

e) Let $I = I_\Phi$. Then $X/I$ is isomorphic to the $K$-linear $C_\infty^\Phi(S)$ of functions that are traces on $\Phi$ of functions from $C_\infty(S)$. It is clear that $C_\infty(\Phi) \subset C_\infty^\Phi(S)$.

A (K)-linear (C_\infty^\Phi(S)) may fail to be Archimedean if (and only if) the set of points at infinity of some (y \in C_\infty^\Phi(S)) (it is always (G_\delta) in (\Phi)) is not an n.n.p. in (\Phi). But the collection of all sets of points at infinity of functions from (C_\infty^\Phi(S)) is precisely the collection of traces on (\Phi) of all z.n.n.p. sets in (S) of type (G_\delta) (each such set in (S) is the set of points at infinity of a function from (C_\infty(S))). This implies d).

Let us note that from the proof it is easy to derive that (\Phi) is a (P)-set if and only if (C_\infty(\Phi)=C_\infty^\Phi(S)).

We omit the straightforward proof of e), observing only that d) and e) are equivalent for any completely disconnected bicompactum.

1. We now translate the second part of Theorem 0 into topological language and obtain from it some corollaries.

Theorem 3. A (P)-set (F) of a quasi-extremal bicompactum is itself quasi-extremal.

Proof. Let (I=I_F). Then (X/I=C_\infty^F(S)), and, by the remark made in the proof of Theorem 2d), (X/I=C_\infty(F)). Since (F) is a (P)-set, (X/I) is a (K_\sigma)-space. Hence (F) is quasi-extremal.

Theorem 4. If (S) is a quasi-extremal bicompactum, (F) is a (P)-set in (S), and (\Phi) is a (P)-set in (F), then (\Phi) is a (P)-set also in (S).

Proof. Obviously,
[
X/I=C_\infty(S)/I_\Phi=C_\infty(\Phi)/I_\Phi^F=C_\infty(\Phi).
]
Here (I_\Phi^F) is the (l)-ideal of the functions (I_\Phi), but considered in (C_\infty(F)), and the last equality follows from the fact that (\Phi) is a (P)-set in (F). By Theorem 3, (\Phi) is quasi-extremal; then (C_\infty(\Phi)) is a (K_\sigma)-space. By Theorem 1, (\Phi) is a (P)-set in (S).

Let us consider one application of the last result to extremally disconnected bicompacta. It is known that in an extremally disconnected bicompactum (Q) there are nonisolated (P)-points if and only if the cardinality of its type is measurable. An obvious consequence of Theorem 3 is the following statement.

Corollary 1. If (Q) is an extremally disconnected bicompactum whose type cardinality (number of cellularity) is nonmeasurable and in which there are no isolated points, and (F) is a (P)-set in (Q), then the bicompactum (F) has no (P)-points (and, in particular, no isolated points).

If the question of the existence of nontrivial quasi-extremal (P)-sets in a quasi-extremal bicompactum is completely solved by Theorems 2a) and 3, then the question of the existence of nontrivial extremally disconnected (P)-sets in an extremally disconnected bicompactum (Q) is considerably more difficult. This question was formulated by R. Sikorski ((^7)). It is equivalent to an old question in the theory of partially ordered spaces. Namely, can one find, in an extended (K)-space, an (l)-ideal (I) which is not a component, for which (X/I) is an (extended) (K)-space? Of course, if the type cardinality of (Q) is measurable, then (Q) even has nontrivial (P)-points. However, it is not known whether there exists a (Q) of nonmeasurable type cardinality having a nontrivial extremally disconnected (P)-set.

Leningrad Institute
of Textile and Light Industry
named after S. M. Kirov

Received
16 I 1970

CITED LITERATURE

1. P. S. Aleksandrov, UMN, 19, No. 6, 3 (1964).
2. A. I. Veksler, Scientific Notes of the Leningrad State Pedagogical Institute named after A. I. Herzen, 183, 107 (1958).
3. B. Z. Vulikh, Introduction to the Theory of Partially Ordered Spaces, 1961.
4. L. Gillman, M. Jerison, Rings of Continuous Functions, 1960.
5. L. Gillman, M. Henriksen, Trans. Am. Math. Soc., 77, No. 2, 340 (1954).
6. Z. T. Dikanova, Siberian Mathematical Journal, 9, No. 4, 804 (1968).
7. R. I. Sikorski, Proceedings of the IV All-Union Topological Conference, 1967, p. 170.