MATHEMATICS

I. I. Parovichenko

ON TOPOLOGICAL SPACES WHOSE WEIGHT IS GREATER THAN THEIR CARDINALITY

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

P. S. Urysohn constructed an example of a \(T_5\)-space (with a single non-isolated point) of cardinality \(\aleph_0\) and weight \(>\aleph_0\) ((\(^{1}\)), p. 206). Subsequently M. Bebutov and V. Schneider constructed an example of a \(T_2\)- (but not \(T_3\)!) space of cardinality \(\aleph_0\), the weight at each point of which is greater than the cardinality (\(^{2}\)). If one takes an \(R\)-minimal Hewitt–Katětov space (\(^{3}\)), obtained, for example, from the space of rational numbers, then one easily obtains an existence theorem for countable \(T_5\)-spaces (cf. 1.2 and 1.3 of (\(^{6}\))) whose weight at each point is greater than the cardinality. However, this still does not give an individual example of such a space, since (and this is acknowledged by Katětov (\(^{3}\))) it is not possible to construct an individual \(R\)-minimal space. Moreover, an existence theorem for analogous uncountable spaces cannot be proved by means of \(R\)-minimal spaces.

The present paper is devoted to the following questions (we use the terminology and notation of (\(^{6}\)); in particular, \(\aleph_\alpha\) denotes a regular, \(\aleph_\lambda\) an irregular, and \(\mathfrak m\) an arbitrary cardinal number): a) enumeration of the cardinality of all \(T_5\)-spaces of cardinality \(\mathfrak m\) and weight \(>\mathfrak m\); b) construction of individual examples of such spaces; c) construction of an individual \(T_5\)-(\(T_3\)-)space of regular (irregular) cardinality \(\aleph_\alpha\) (\(\aleph_\lambda\)), the weight at each point of which is greater than the cardinality; d) construction of a completely simple countable \(T_2\)-(\(T_5\)-)space with uncountable weight at each (at one) point, and also of a countable connected \(T_2\)-space with uncountable weight at each point; e) a negative solution of the problem of \([ab]\)-compactness (for irregular \(\alpha\)); the solution of it undertaken in (\(^{4}\)) contains, as P. S. Aleksandrov kindly informed me, a gap (the inevitability of which is now also becoming clear).

Since, in view of (\(^{6}\)), the set of all \(T_5\)-spaces of cardinality \(\mathfrak m\) has cardinality \(2^{2^{\mathfrak m}}\), while the set of all \(T\)-spaces of cardinality and weight \(\leq \mathfrak m\) has cardinality \(2^{\mathfrak m}\), it follows that the cardinality of the set of all \(T_5\)-spaces of cardinality \(\mathfrak m\) and weight \(>\mathfrak m\) is \(2^{2^{\mathfrak m}}\)—greater than the cardinality of all spaces of cardinality and weight \(\leq \mathfrak m\), and a) is accomplished.

Let \(\mathfrak S_\lambda=\sum_{\nu<\lambda}^{(A)}\Omega_{\nu+1}\) and \(\mathfrak P_\alpha=\Omega_\alpha \times \Omega_\alpha\) (\(^{6}\)).

Lemma 1. \(\mathfrak P_\alpha\) is a \(T_1^\alpha\overline{T}_2^+\)-\((\aleph_\alpha)\)-space, whose local weight is \(>\aleph_\alpha\) (cf. 2.2 of (\(^{6}\))), and \(\mathfrak S_\lambda\) is a \(T_1\overline{T}_2^+\)-\((\aleph_\lambda)\)-space, whose local weight is \(>\aleph_\lambda\).

The first part of the lemma is proved by contradiction with the help of 1.2 and 2.8 of (\(^{6}\)) and the fact that in every set \(P\subseteq\mathfrak P_\alpha\) there is contained a set equipotent to \(P\) and closed in itself.

We shall prove the second part of the lemma. It is clear that for an arbitrary point \(x_0\in\mathfrak S_\lambda\) the pseudo-weight \(\psi(x_0)=\aleph_\lambda\) (cf. 2.7 of (\(^{6}\))), so that the weight \(\chi(x_0)\geq \psi(x_0)=\)

\(= \aleph_\lambda\), and it remains for us to show that \(\chi(x_0) \geq \aleph_\lambda\), which we shall do by contradiction. Let \(\chi(x_0) = \aleph_\lambda\). Thus, let \(\mathfrak{G}(x_0)=\{G^\xi(x_0)\}_{\xi<\omega_\lambda}\) be an open base \(\mathfrak{S}_\lambda\) at \(x_0\), where \(G_\xi(x_0)=\bigcap_{\nu<\lambda}G_{\nu+1}^\xi\) and \(G_{\nu+1}^\xi\) \((\xi<\omega_\lambda)\) are nonempty and open in \(\Omega_{\nu+1}\), with \(x_0\in G_{\nu_0+1}^\xi\) \((\xi<\omega_\lambda)\). Put
\[ G_{\nu+1}=\left(\bigcap_{\xi\leq\omega_\nu}G_{\nu+1}^\xi\right)\setminus y_{\nu+1}, \]
where \(y_{\nu+1}\ne x_0\) and \(y_{\nu+1}\) belongs to the indicated intersection. Clearly,
\[ G(x_0)=\bigcup_{\nu<\lambda}G_{\nu+1} \]
is a neighborhood of \(x_0\) in \(\mathfrak{S}_\lambda\). We take in the base \(\mathfrak{G}(x_0)\) a neighborhood \(G^{\xi_0}(x_0)\subset G(x_0)\); then \(G_{\nu+1}^{\xi_0}\subset G_{\nu+1}\) for every \(\nu<\lambda\), and from the definition of \(G_{\nu+1}\) we have \(\xi_0>\omega_\nu\) for every \(\nu<\lambda\), i.e. \(\xi_0\geq\omega_\lambda\). But \(\xi_0<\omega_\lambda\), and we obtain the required contradiction.

Definition. Let \(\mathfrak{M}\) be a \(T\)-space and \(x_0\in\mathfrak{M}\). By \(x_0(\mathfrak{M})\) we shall agree to denote the space obtained from \(\mathfrak{M}\) by weakening the topology by declaring all points of \(\mathfrak{M}\), except \(x_0\), isolated.

Lemma 2. If \(\mathfrak{M}\) is a \(T_1\)-space and \(x_0\in\mathfrak{M}\), then \(x_0(\mathfrak{M})\) is a \(T_5\)-space, and the weights of \(\mathfrak{M}\) and \(x_0(\mathfrak{M})\) at \(x_0\) coincide.

Now the assertion required for b) follows at once from the lemmas given above. Let
\[ M_\omega=\bigcup_{0\leq n<\omega}M_n, \]
where the \(M_n\) are pairwise disjoint sets, each of cardinality \(\mathfrak{m}\), and let \(\varphi\) be a one-to-one mapping of \(M_\omega\setminus M_0\) onto \(M_\omega\), subject to the conditions: 1) \(\varphi(M_{n+1})=M_n\) \((n=0,1,2,\ldots)\), and 2) \(\varphi^{-1}(x_n)=M_{n+1}^{x_n}\) has cardinality \(\mathfrak{m}\) \((x_n\in M_n)\); let \(\mathfrak{M}_{n+1}^{x_n}\) be some \(T_1T_2^{1}\)-space defined on \(M_{n+1}^{x_n}\). We shall then agree to associate with the set \(M_\omega\) the space \(\mathfrak{M}_\omega\), defined on \(M_\omega\) by the neighborhoods:
\[ U(x_n)=x_n\cup G_{n+1}^{x_n}\cup\left\{\bigcup_{i<\omega}\varphi^{-i}\left(G_{n+1}^{x_n}\right)\right\}, \]
where \(x_n\in M_n\), \(G_{n+1}^{x_n}\) is an arbitrary nonempty open set in \(\mathfrak{M}_{n+1}^{x_n}\), and \(\varphi^{-i}\) are defined inductively: \(\varphi^{-2}=\varphi^{-1}\varphi^{-1}\), etc. \(\mathfrak{M}_\omega\) is zero-dimensional and therefore is always a \(T_3\)-space.

Lemma 3. The weight of \(\mathfrak{M}_\omega\) at \(x_n\) is not less than the weight of the space \(\mathfrak{M}_{n+1}^{x_n}\).

Proof. We shall regard \(M_{n+1}^{x_n}\cup x_n\) as a subspace in \(\mathfrak{M}_\omega\); in this subspace all points of \(M_{n+1}^{x_n}\) are isolated, and it is homeomorphic to the space \(x_n(\mathfrak{M}_{n+1}^{x_n}\cup x_n)\), where in the parentheses stands the space obtained from \(\mathfrak{M}_{n+1}^{x_n}\) by adjoining the point \(x_n\) in the manner 2,3 of \((^6)\). In view of 2,3 of \((^6)\) and Lemma 2, the weight of \(M_{n+1}^{x_n}\cup x_n\subset\mathfrak{M}_\omega\) at \(x_n\) is equal to the weight of \(\mathfrak{M}_{n+1}^{x_n}\), and hence the weight of \(\mathfrak{M}_\omega\) at \(x_n\) is not less than the weight of \(\mathfrak{M}_{n+1}^{x_n}\), as was required to prove.

Putting now identically \(\mathfrak{M}_{n+1}^{x_n}\equiv\mathfrak{S}_\lambda\), or \(\equiv\mathfrak{P}_\alpha\), we obtain a \(T_3\)-space \(\mathfrak{M}_\omega\) whose weight at each point is greater than its cardinality. If \(\mathfrak{m}=\aleph_\alpha\) and \(\mathfrak{M}_{n+1}^{x_n}\equiv\mathfrak{P}_\alpha\), then, since \(\mathfrak{M}_\omega\) is then an \(\aleph_{\alpha+1}\)-bicompact \(T_3\)-space, in view of 1,5 of \((^6)\) \(\mathfrak{M}_\omega\) is a \(T_4\)-space, and since all properties used for the proof of the axiom \(T_4\) are hereditary, \(\mathfrak{M}_\omega\) is a \(T_5\)-space, and c) is accomplished.

Remark. If \(\mathfrak{m}=\aleph_0\) and \(\mathfrak{M}_{n+1}^{x_n}\equiv\Omega_0\) \((^6)\), then \(\mathfrak{M}_\omega\) is homeomorphic to the space of rational numbers \(R\), so that our space \(\mathfrak{M}_\omega\) in the general case is obtained by weakening the topology in \(R\) (since every countable \(T_1\)-space is obtained by weakening the topology of \(\Omega_0\)).

Let (as above) \(R\) be the space of rational numbers with the usual topology; by \(\widetilde{R}\) denote the space obtained from \(R\) by weakening the topology according to the rule: as closed sets in \(\widetilde{R}\) one regards all sets of the form \(F\cup N\), where \(F\) is closed and \(N\) is nowhere dense in \(R\).

It is quite easy to see that \(\widetilde R\) is an \(F_2\)-space with uncountable weight at each point; the space \(x_0(\widetilde R)\), for any \(x_0 \in \widetilde R\), is, by elementary lemma 2, a countable \(T_5\)-space with uncountable weight, simpler than the space of P. S. Urysohn \({}^{1}\). Further, if at the vertices of the lower bases of the triangles that serve Bing \({}^{5}\) for constructing a countable connected \(T_2\)-space, instead of the sets \(\delta_1 \cap R\) and \(\delta_2 \cap R\) (\(\delta_1\) and \(\delta_2\) are intervals on the abscissa axis) one takes the sets \(\delta_1 \cap (R \setminus N_1)\) and \(\delta_2 \cap (R \setminus N_2)\), where \(N_1\) and \(N_2\) are nowhere dense in \(R\), then we obtain a countable connected \(T_2\)-space with uncountable weight at every point, and d) is realized.

Finally, to realize e), we shall show that an isolated space \(\mathfrak E_\lambda\) of irregular cardinality \(\aleph_\nu\) is an \([\aleph_\lambda,\aleph_{\lambda+1}]\)-compact space for which the property \(B^{0}_{\aleph_\lambda,\aleph_{\lambda+1}}\) \({}^{2}\) is not fulfilled, even if for it one considers covers not designated, i.e. with pairwise distinct elements. Let \(f\) be a one-to-one mapping of \(\mathfrak E_\lambda\) onto \(\mathfrak S_\lambda\); take in \(\mathfrak E_\lambda\) a cover \(\mathfrak B_{\lambda+1}\) of cardinality \(\aleph_{\lambda+1}\), into which enter all one-point and some other sets that are carried by means of \(f\) into proper closed subsets of \(\mathfrak S_\lambda\) (they may be regarded as pairwise distinct, since the weight of \(\mathfrak S_\lambda\) is greater than \(\aleph_\lambda\)). Since \(\mathfrak S_\lambda\) is a \(\overline T_2(\aleph_\lambda)\)-space, no subcover of cardinality \(<\aleph_\lambda\) can be selected from \(\mathfrak B_{\lambda+1}\), as was required. But the required \(\mathfrak B_{\lambda+1}\) can be constructed still more simply: it is enough to represent \(\mathfrak E_\lambda\) as the union of disjoint sets \(\mathfrak E_\lambda=\mathfrak E_\lambda^{0}\cup \mathfrak E_\lambda^{1}\), each of cardinality \(\aleph_\lambda\); then as \(\mathfrak B_{\lambda+1}\) one may take the union \(\mathfrak E_\lambda^{0}\cup \widetilde{\mathfrak E}_\lambda^{1}\), where \(\widetilde{\mathfrak E}_\lambda^{1}\) is a cover of \(\mathfrak E_\lambda^{1}\) of cardinality \(\aleph_{\lambda+1}\).

Kishinev State
University

Received
12 III 1957

References

\({}^{1}\) P. S. Urysohn, Works on Topology, Moscow—Leningrad, 1951.
\({}^{2}\) M. Bebutov, V. Shneider, Scientific Notes of Moscow State University, issue 13, Math., book 3, 157 (1939).
\({}^{3}\) M. Katetov, Math. Sbornik, 21 (63), No. 1, 3 (1947).
\({}^{4}\) Yu. Smirnov, Izv. AN SSSR, ser. math., 14, 155 (1950).
\({}^{5}\) R. Bing, Proc. Am. Math. Soc., 4, No. 3, 474 (1953).
\({}^{6}\) I. I. Parovichenko, DAN, 115, No. 5 (1957).