MATHEMATICS

I. I. Parovichenko

ON SOME SPECIAL CLASSES OF TOPOLOGICAL SPACES AND (\delta s)-OPERATIONS

(Presented by Academician P. S. Aleksandrov on 13 III 1957)

Below (\aleph_\chi) is a regular, (\aleph_\lambda) an irregular, and (\mathfrak m) an arbitrary infinite cardinal number. If (\Sigma_1) and (\Sigma_2) are certain classes of (T)-spaces, then by a (\Sigma_1\Sigma_2)-space we shall mean a space of the class (\Sigma_1\cap\Sigma_2).

1. We shall call a (T)-space (\mathfrak M) a (T^\alpha)-space if the intersection of every system of cardinality (<\aleph_\chi) of open sets in (\mathfrak M) is open in (\mathfrak M). An example of a (T^\alpha)-space (which is not a (T^{\alpha+1})-space) is the ordered set of Hausdorff normal type (\eta_\alpha) ((^1)) with the natural interval topology; another example is the space (\Omega_\alpha) of cardinality (\aleph_\chi), all proper closed sets of which are all subsets of cardinality (<\aleph_\alpha). But a (T^\alpha)-space can be constructed from an arbitrary space (\mathfrak M) by passing to the space (T^\alpha\mathfrak M), whose points are the same as in (\mathfrak M), while the topology is determined by the open base consisting of all possible intersections of cardinality (<\aleph_\chi) of open sets of (\mathfrak M) (or of elements of some open base of (\mathfrak M)). It is easy to see that (T_\alpha(T^\alpha\mathfrak M)=T\mathfrak M^\chi), and (\mathfrak M) is a (T^\alpha)-space if and only if (\mathfrak M=T^\alpha\mathfrak M). Put (T^\alpha T_i=T_i^\alpha) ((i=0,1,2,\ldots,5)).

1.1. At every point of a (T_1^\alpha)-space the weight (and pseudo-weight) is either equal to (1), or (\ge \aleph_\chi).

1.2. At every point (x) of a (T^\alpha)-space whose weight is equal to (\aleph_\alpha), the character weight is defined, and it is also equal to (\aleph_\chi).

1.3. For (\alpha>0), and for countable spaces also for (\alpha=0), a (T_0^\alpha)-space is regular if and only if it is zero-dimensional.

We shall call a (T)-space (\mathfrak M) (\mathfrak m)-bicompact if from every open covering of (\mathfrak M) one can select a subcovering of cardinality (<\mathfrak m).

Analogously to ((^2)), p. 147, and ((^7)), the following propositions are proved:

1.4. In order that a (T_3^\alpha)-space be (\aleph_\alpha)-bicompact, it is necessary and sufficient that it be closed in every (T_3^\alpha)-space topologically containing it.

1.5. If a (T_3^\alpha)-space (\mathfrak M) is (\aleph_{\chi+1})-bicompact, then (\mathfrak M) is a (T_4^\alpha)-space.

1.6. A continuous image of an (\mathfrak m)-bicompact space is an (\mathfrak m)-bicompact space.

The (T^\alpha)-product of (T)-spaces (\mathfrak M_\xi) ((\xi\in\Xi)), denoted by

[
\prod_{\xi\in\Xi}^{(\alpha)} \mathfrak M_\xi,
]

is the product of the spaces (T^\alpha\mathfrak M_\xi), differing from the Tychonoff product only in that, in specifying elementary open sets (cf. ((^5))), instead of a system of indices of cardinality (<\aleph_0), one fixes a sy-

system of cardinality (<\aleph_x). It is easy to see that
(T^\alpha\left(\prod \mathfrak{M}\xi\right)=\prod^{(\alpha)}\mathfrak{M}\xi), where (\prod) is the Tikhonov product.

1.7. For (i=0,1,2,3), the class of (T_i^\alpha)-spaces is hereditary and (T^\alpha)-multiplicative.

2. Consider the following two correspondences:

A. Let (M\cup x) be a (T_1)-space in which (x) is the only nonisolated point. To it one may assign the space (\mathfrak{M}), defined on (M), all of whose nonempty open sets are all sets of the form (U(x)\cap M), where (U(x)) is a neighborhood of (x) in the original space.

B. Let (\Phi) be an (\mathfrak{m})-(\delta s)-operation ((²), p. 91 and (⁶)) with index set (M) of cardinality (\mathfrak{m}). It is easy to see that the base (N_\Phi) (²) for an arbitrary such operation (\Phi) is nothing other than a system (N) of subsets of (M), subject to two conditions: 1) (\Lambda\in N) and 2) (U\in N\ \&\ U'\supset U\to (U'\in N)); however, some operations (for example, the lower limit) have the following two additional properties: 3) for every (n\in U\in N\to (U\setminus n\in N)) and 4) ((U_1\in N\ \&\ U_2\in N)\to (U_1\cap U_2\in N)). These latter operations we shall call lower. To them we also assign the space (\mathfrak{M}), defined on (M) by the family (N_\Phi) as the collection of all nonempty open sets. Then the base (N_{\overline{\Phi}}) of the complementary operation (\overline{\Phi}) is the family of all everywhere dense subsets of (\mathfrak{M}).

In items A and B we have arrived at a (T_1)-space (\mathfrak{M}) having the properties:

(\alpha)) any neighborhoods of any two points intersect;

(\beta)) if (G) is nonempty and open and (U\supset G), then (U) is open.

It is easy to see that (T_1)-spaces with properties (\alpha)) and (\beta)), in the sense of our correspondence, correspond one-to-one to spaces (M\cup x) from A and to lower operations from B. For example, the space (\Omega_0) (see above) corresponds to an ordered space of type (\omega+1) and to the lower limit.

(T)-spaces with property (\alpha)) we shall call anti-Hausdorff, or (\overline{T}_2)-spaces, and (\overline{T}_2)-spaces with property (\beta))—(\overline{T}_2^{+})-spaces.

Introduce further axioms:

(\overline{T}_2(\mathfrak{m})): in a (T)-space the intersection of any system of cardinality (<\mathfrak{m}) of nonempty open sets is nonempty.

(\overline{T}_2^{+}(\mathfrak{M})): in a (T)-space, (\overline{T}_2(\mathfrak{M})) and (\beta)) hold.

The space (\Omega_\alpha) serves as an example of a (T_1^\alpha\overline{T}_2^{+}(\aleph_x))-space.

2.1. If (F_1) and (F_2) are equipotent proper closed sets in a (T_1\overline{T}_2^{+})-space (\mathfrak{M}), then a one-to-one mapping of (\mathfrak{M}) onto itself under which (F_1) is mapped onto (F_2), while all points of (\mathfrak{M}\setminus(F_1\cup F_2)) remain fixed, is a homeomorphism.

2.2. In a (T_1\overline{T}_2^{+})-space, topologically indistinguishable points, in particular, have the same, respectively, pseudo-weight and weight; moreover, the local weight of the space is equal to its integral weight.

2.3. If to a (T_1\overline{T}_2^{+})-space (\mathfrak{M}) one adjoins a point (x) with neighborhoods (U(x)=G\cup x), where (G) is an arbitrary nonempty open set of (\mathfrak{M}), then the resulting space is homeomorphic to (\mathfrak{M}).

2.4. If in correspondence A, (M\cup x) is a (T_1^\alpha)-space, then the corresponding space (\mathfrak{M}) is a (T_1^\alpha\overline{T}_2^{+}(\aleph_x))-space, and conversely.

2.5. An (\mathfrak{m})-(\delta s)-operation (\Phi) is a lower operation coinciding with its complementary operation if and only if (N_\Phi) (²) is a maximal centered system of subsets of the index set (M) with empty intersection.

Remark. If the last intersection is nonempty, and then it consists of a single element (n_0\in M), then such an (N_\Phi) will also determine an (m)-(\delta s)-operation coinciding with its complementary one, namely the operation (\Phi(E_n)=E_n), ((n\in M)); however, for this (\Phi) property 3) is not fulfilled, and the corresponding space would be a (T_0\overline{T}_{2}^{+})-space with a single nonclosed point ((n_0)).

2.6. The sets of all (T^{-})-, (T_1\overline{T}_{2}^{+})- and (T_5)-spaces of cardinality (m), and also the set of all (m)-(\delta s)-operations (in particular, of lower operations and of those coinciding with their complementary operations), have cardinality (2^{2^m}).

Indeed, let (M) be an isolated space of cardinality (m); in view of ((^4)), the cardinality (\beta M=2^{2^m}). Let (M\cup x) be a subspace in (\beta M), where (x\in\beta M\setminus M); we divide the set of all such spaces into equivalence classes, where homeomorphism is taken as the equivalence relation. Since one class contains a set of spaces of cardinality (\leq mm=2^m<2^{2^m}), there will be (2^{2^m}) classes in all; whence, taking account of the preceding result and ((^5)), we obtain the desired result.

Let (\mathfrak X_\xi) ((\xi\in\Xi)), (\mathfrak M), and (\mathfrak N) be (T_1\overline{T}{2}^{+})-spaces, where the (\mathfrak X\xi) are pairwise disjoint. We shall call the union (\bigcup_{\xi\in\Xi}\mathfrak X_\xi) with a topology in which the open sets are (\Lambda) and all sets of the form (G=\bigcup_{\xi\in\Xi}G_\xi), where (G_\xi) are nonempty open sets in the corresponding (\mathfrak X_\xi), an (A)-sum

[
\sum_{\xi\in\Xi}^{(A)}\mathfrak X_\xi;
]

we call the abstract product of the corresponding point sets, with the topology in which the open sets are (\Lambda) and every set (W) such that there exists (W_0\subseteq W),

[
W_0=\bigcup_{x\in U}[x,V_x],
]

where (U) is a nonempty open set in (\mathfrak M), (V_x) are nonempty open sets in (\mathfrak N), and ([x,V_x]) is the set of all pairs ((x,y)), (y\in V_x), the (A)-product

[
\mathfrak M \stackrel{A}{\times} \mathfrak N .
]

2.7. If (\mathfrak X_\xi) is a (T_1\overline{T}{2}^{+}(m\xi))-space ((\xi\in\Xi)), then (\sum_{\xi\in\Xi}^{(A)}\mathfrak X_\xi) is a (T_1\overline{T}{2}^{+}(\sup m_\xi))-space, and each summand is everywhere dense in the (A)-sum.

2.8. The (A)-product of two (T_1^{\alpha}\overline{T}{2}^{+}(\mathfrak N\alpha))-spaces is a (T_1^{\alpha}\overline{T}{2}^{+}(\mathfrak N\alpha))-space.

Kishinev State
University

Received
12 III 1957

CITED LITERATURE

1. F. Hausdorff, Grundzüge der Mengenlehre, N. Y., 1949, p. 179.
2. Ф. Хаусдорф, Теория множеств, М.—Л., 1937.
3. B. Pospíšil, Ann. of Math., 38, no. 4, 845 (1937).
4. П. Александров, Усп. матем. наук, 11, no. 1 (17), 26 (1947).
5. П. Александров, Введение в общую теорию множеств и функций, М.—Л., 1948, p. 392.
6. В. Шнейдер, Уч. зап. МГУ, vol. 135, mathematics, 2, 76 (1948).
7. Ю. Смирнов, Изв. АН СССР, сер. матем., 14, 165 (1950).