UDC 513.83

MATHEMATICS

N. S. LASHNEV

ON CONTINUOUS PARTITIONS OF COLLECTIVELY NORMAL SPACES

(Presented by Academician P. S. Aleksandrov on 17 I 1969)

Definition 1. Let there be a set \(L\) of cardinality \(\tau\), and let to each element \(\lambda\) of \(L\) there be assigned a discrete system of sets \(\gamma_\lambda\). Let

\[ \gamma=\bigcup_{\lambda\in L}\gamma_\lambda . \]

We shall say that the system \(\gamma\) is \(\tau\)-discrete.

Definition 2. Let \(X\) be a completely regular space. We shall say that its uniform weight is equal to \(\tau\) if there exists a uniform structure with a base of \(\tau\) covers, compatible with the topology of the space \(X\), and for any cardinal number \(\sigma<\tau\) no such uniform structure exists.

Theorem. Let \(X\) be a collectively normal space of uniform weight \(\tau\), and let \(\{A\}\) be its continuous partition. Then in all elements of the partition, except for the elements of some \(\tau\)-discrete system, every discrete set has cardinality \(<\tau\).

Proof. 1. Denote by \(\{\alpha_\lambda\}_{\lambda\in L}\) a base of the uniform structure in \(X\), consisting of \(\tau\) covers \(\alpha_\lambda\). The system \(\{\alpha_\lambda\}\) has the following properties: a) for any two covers \(\alpha_{\lambda_1}\) and \(\alpha_{\lambda_2}\) there exists a third, which is inscribed in their intersection \(\alpha_{\lambda_3}<\alpha_{\lambda_1}\wedge\alpha_{\lambda_2}\); b) if \(x\) is a point in the space \(X\), then the system of all stars of this point with respect to the covers \(\alpha_\lambda\) forms a base at the point \(x\).

1. To each \(\lambda\) from the set \(L\) we assign a discrete system of elements of the partition \(\{A\}\). For this purpose denote by \(V_\lambda(A)\) the star of the element \(A\) with respect to the cover \(\alpha_\lambda\). Into the system \(\gamma_\lambda\) we select those and only those elements of the partition which have the property: the star of the element of the partition with respect to the system \(\alpha_\lambda\) contains wholly no element of the partition distinct from it itself,

\[ A_0\in\alpha_\lambda \Longleftrightarrow V_\lambda(A_0)\cap A_1\ne A_1 \]

for \(A_1\ne A_0\).

The discreteness of the system \(\alpha_\lambda\) follows from the continuity of the partition \(\{A\}\): if \(A_0\notin\gamma_\lambda\), then for any \(x\in A_0\) there exists a neighborhood \(Ox\) which does not meet the bodies \(G_\lambda\) of the system \(\alpha_\lambda\). Otherwise one would have \([G_x]\cap A_0\ne\Lambda\), and by virtue of the continuity of the partition the neighborhood \(V_\lambda(A_0)\) of the set \(A_0\) would contain wholly some element \(A_1\in\alpha_\lambda\), which contradicts the construction of the system \(\alpha_\lambda\).

1. We shall prove that a discrete subset of any element of the partition which has not entered into any system \(\alpha_\lambda\) has cardinality \(<\tau\). Suppose this is not so: \(A_0\notin \bigcup_{\lambda\in L}\gamma_\lambda\) and \(A_0\supset P\), where \(P\) is a discrete set of cardinality \(\tau\). Since \(L\) has cardinality \(\tau\), one can establish a one-to-one correspondence between \(P\) and \(L\); then each point \(x\in P\) acquires an index \(\lambda\): \(P=\{x_\lambda\}_{\lambda\in L}\).

2. Since \(X\) is collectively normal, the discrete system of points \(\{x_\lambda\}\) has a discrete system of neighborhoods \(\{Ox_\lambda\}\) in the space \(X\).

1. For each point \(x_\lambda \in P\) one can choose a cover \(a(x_\lambda) \in \{\alpha_\lambda\}\) so that the star of this cover with respect to the point \(x_\lambda\) is contained in the set \(Ox\). It may happen that the same cover \(\alpha_\lambda\) corresponds to different points \(x_1\) and \(x_2\) of \(P\). We have constructed a mapping \(\varphi\) of the set \(P=\{x_\lambda\}\) into the set of covers \(\{\alpha_\lambda\}_{\lambda \in L}\).

2. In item 3 a one-to-one correspondence between the sets \(L\) and \(P\) was established. It can be turned into a one-to-one mapping \(\psi:\{x_\lambda\}\to\{\alpha_\lambda\}\), by assigning to each point \(x_\lambda\) the cover with the same index.

3. Let us construct a third mapping \(\omega\) of the set \(P\) into the set of covers \(\alpha_\lambda\). Using property b) of item 1, to each point \(x \in P\) we assign some cover \(\alpha_x\) belonging to the system \(\{\alpha_\lambda\}\), in such a way that, for any \(\lambda\), one has \(\alpha_x^\lambda < \alpha(x_\lambda)\cap \alpha_\lambda\), or \(\omega(x)<\varphi(x)\cap \psi(x)\) \((x \in P)\).

4. For any \(x \in P\) we have:

\[ St(\alpha_x,x)\subset Ox; \tag{1} \]

there exists an element \(A_x\) of the decomposition \(\{A\}\) such that

\[ St(\alpha_x,A_x)\supset A. \tag{2} \]

1. Since \(\{\alpha_x\}_{x\in P}\) forms a base of the uniform structure, we have \(\left[\bigcup_{x\in P} A_x\right]\supset A\).

2. Consider an arbitrary \(A_x\). In view of (2), there exists an element \(H\) of the cover \(\alpha_x\) which intersects \(A_x\) and contains the point \(x\). From (1) it follows that \(A_x\cap Ox\ne \Lambda\). Choose in the set \(A_x\cap Ox\) a point \(y(x)\). Obviously, the set \(Q\) of all points \(y(x)\) is discrete.

3. Consider the open set \(X\setminus Q\). Since the decomposition \(\{A\}\) is continuous, there exists an open saturated set \(U\subset X\setminus Q\) which contains \(A_0\). It is easy to see that \(X\setminus Q\) does not contain entirely any one of the sets \(A_x\) \((x\in P)\); hence, for any \(x\in P\), \(A_x\cap U=\Lambda\) and \((\bigcup A_x)\cap U=\Lambda\), which contradicts the fact that \(\left[\bigcup_{x\in P} A_x\right]\supset A_0\). The theorem is proved.

Faculty of Mechanics and Mathematics
M. V. Lomonosov Moscow State University

Received
8 I 1969

REFERENCES

\(^1\) A. V. Arkhangel’skii, DAN, 166, No. 6 (1966). \(^2\) N. S. Lašnev, DAN, 165, No. 4 (1965). \(^3\) V. V. Filippov, DAN, 178, No. 3 (1968).