Reports of the Academy of Sciences of the USSR
1964. Vol. 155, No. 4

MATHEMATICS

K. KURATOWSKI

CHARACTERIZATION OF REGULAR STRUCTURES BY MEANS OF EXPONENTIAL TOPOLOGY

(Presented by Academician P. S. Aleksandrov on 8 I 1964)

1. Let a distributive structure \(\Gamma=(L,\cup,\cap,0,1)\) be given. We shall call it a Urysohn structure \(\left(^{1}\right)\) if

\[ (a\not\subset b)\to \bigvee_c (c\cap a\ne 0)(c\cap b=0) \tag{1} \]

(\(\bigvee_c\) denotes “there exists a \(c\) such that…”).

We shall call the structure \(\Gamma\) regular \(\left(^{2}\right)\) if

\[ (a\not\subset b)\Rightarrow \bigvee_{cd}(c\cup d=1)(a\not\subset c)(b\cap d=0). \tag{2} \]

Denoting by \(I(a)\) and \(J(a)\) the ideals

\[ I(a)=\{x:x\subset a\},\qquad J(a)=\{x:x\cap a=0\}, \]

we shall call the exponential topology for \(L\) the topology whose open subbase consists of the sets \(L\setminus I(a)\) and \(J(a)\), where \(a\in L\). In other words, an open base of the space \(L\) consists of the sets

\[ B(a_0,a_1,\ldots,a_n)=J(a_0)\setminus I(a_1)\setminus\cdots\setminus I(a_n)= \]
\[ =\{x:(x\cap a_0=0)(x\not\subset a_1)\cdots(x\not\subset a_n)\}. \tag{3} \]

1. Theorem. Let \(\Gamma\) be a Urysohn structure. In order that the set \(F=\{(x,y):x\subset y\}\) be closed in the space \(L\times L\), it is necessary and sufficient that \(\Gamma\) be a regular structure.

Proof. \(1^\circ\). The condition is sufficient. First of all, it is easy to see that in formula (2) the arrow \(\Rightarrow\) may be replaced by the equivalence sign \(\equiv\). This makes it possible to obtain the formula

\[ L^2\setminus F=\bigcup_{c\cup d=1}[L\setminus I(c)]\times J(d). \]

Consequently, the set \(L\setminus F\) is open.

\(2^\circ\). The condition is necessary.\(**\) Let the set \(F\) be closed and let \(a\not\subset b\). It is required to determine \(c\) and \(d\) in such a way that

\[ c\cup d=1,\qquad a\not\subset c,\qquad b\cap d=0. \tag{4} \]

Since the point \((a,b)\) belongs to the open set \(L^2\setminus F\), there exist two sets \(U\) and \(V\), belonging to the open base of the space \(L\), such that

\[ a\in U,\qquad b\in V,\qquad U\times V\subset L^2\setminus F. \]

* This topology coincides with the topology of the structure \(2^X\) of all closed sets of the topological space \(X\) (see \((^{3-5,7})\)). Ponomarev denotes it by \(\psi X\) \((^{6})\). It is easy to verify that a topological \(T_1\)-space \(X\) is regular if and only if the structure \(2^X\) is regular in the sense of formula (2).

** The problem of the necessity of this condition was posed by me in a talk at Moscow State University on 6 XII 1963.

In other words (see (3)): there exist two finite systems \(a_0, a_1,\ldots,a_m\) and \(b_0,b_1,\ldots,b_n\) such that:

\[ a\cap a_0=0,\quad a\not\subset a_i\ (i=1,\ldots,m),\quad b\cap b_0=0,\quad b\not\subset b_j\ (j=1,\ldots,n), \tag{5} \]

\[ [(x\cap a_0=0)(x\not\subset a_i)(y\cap b_0=0)(y\not\subset b_j)]\Rightarrow ((x,y)\in L^2\setminus F)\equiv (x\not\subset y). \tag{6} \]

Let \(d=b_0\). We shall prove that among the numbers \(1,\ldots,m\) there exists an \(i\) for which \(a_0\cup a_i\cup b_0=1\); thus, taking \(c=a_0\cup a_i\), condition (4) will be proved (on the basis of the first three parts of formula (5)).

Suppose that this is not so; in other words, \(a_0\cup a_i\cup b_0\ne 1\) for \(i=1,\ldots,m\). Substitute in formula (1) \(a=1\) and \(b=a_0\cup a_i\cup b_0\); consequently, there exists \(c_i\) such that

\[ c_i\ne 0,\quad c_i\cap(a_0\cup a_i\cup b_0)=0. \tag{7} \]

Let \(x=c_1\cup\cdots\cup c_m\) and \(y=x\cup b\). It is easy to verify (on the basis of (7) and the second half of (5)) that \(x\) and \(y\) satisfy all the conditions of formula (6) (enclosed in the brackets \([\ ]\)). But then, according to formula (6), we get \(x\not\subset y\), which clearly contradicts the definition of \(y\).

1. A structure \(\Gamma\) is called a Brouwer algebra if in it there exists a difference \(x-y\) satisfying the following condition*:

\[ (x-y\subset z)\equiv [x\subset (y\cup z)]. \]

It is easy to see that if \(\Gamma\) is a regular Brouwer algebra, then the sets \(\{(x,y): x-y=0\}\) and \(\{(x,y): x-y\subset a\}\) are closed (in \(L\times L\)). In other words (see (?)), the mapping \(x-y\) (from \(L\times L\) into \(L\)) is lower semicontinuous.

From the theorem proved by us we obtain

Corollary. Let \(\Gamma\) be a Wallman structure and a Brouwer algebra.

Then the following propositions are equivalent:

\(1^\circ\). \(\Gamma\) is a regular structure.

\(2^\circ\). The set \(\{(x,y): x-y=0\}\) is closed.

\(3^\circ\). The operation \(x-y\) is lower semicontinuous.

Recall that for Wallman structures \(\Gamma\) it has been proved (see \((^2)\)) that the following propositions are equivalent:

\(\alpha)\) \(\Gamma\) is a normal structure;

\(\beta)\) the set \(\{(x,y): x\cap y=0\}\) is open;

\(\gamma)\) the operation \(x\cap y\) is upper semicontinuous,

where a normal structure is a structure subject to the condition:

\[ (a\cap b=0)\Rightarrow \bigvee_{cd}(c\cup d=1)(a\cap c=0=b\cap d), \]

and upper semicontinuity of the mapping \(x\cap y\) means that the set \(\{(x,y): x\cap y\cap a=0\}\) is open for every \(a\in L\).

Warsaw University
Warsaw, Polish People’s Republic

Received
8 I 1964

REFERENCES

\(^{1}\) H. Wallman, Ann. Math., 39, 112 (1938).
\(^{2}\) K. Kuratowski, Bull. Acad. Polon., 11 (1963).
\(^{3}\) L. Vietoris, Monatsh. f. Math., 31 (1921).
\(^{4}\) O. Frik, Trans. Am. Math. Soc., 51, 569 (1942).
\(^{5}\) E. Michael, ibid., 71, 152 (1951).
\(^{6}\) V. Ponomarev, Matem. sborn., 48, No. 2, 191 (1959).
\(^{7}\) K. Kuratowski, Rendicont. di Mat., Roma, 22 (1963).
\(^{8}\) R. Engelking, Bull. Acad. Polon., 11 (1963).

* In the case of the structure \(2^X\), the difference denotes \(\overline{A-B}\).

** In the special case of the structure \(2^X\), where \(X\) is a topological \(T_1\)-space, this proposition was proved by Engelking \((^8)\).