I. M. DEKTYAREV

A CLOSED-GRAPH THEOREM FOR ULTRACOMPLETE SPACES

(Presented by Academician P. S. Aleksandrov on 13 III 1964)

The paper proves a closed-graph theorem for multivalued mappings of ultracomplete spaces. Theorems of this kind for linear mappings of locally convex spaces were established by V. Pták (¹) and J. L. Kelley (²). V. L. Levin and D. A. Raikov (³) proved analogous theorems in the case of so-called biregular mappings, for one class of uniform spaces (including ultracomplete spaces) which they called \(B\)-complete.

Let us introduce some notation that will be useful below. Let \(X\) and \(Y\) be uniform spaces (not assumed to be separated). We shall regard any subset of the product \(X \times Y\) as the graph of a multivalued mapping from \(X\) into \(Y\), and denote it by the same letter as this mapping. In particular, entourages in \(X\) are regarded as multivalued mappings of the space \(X\) into itself.

By \(\{U_i\}_{i \in I}\) we denote an arbitrary fundamental system of symmetric entourages in \(X\), and by \(\{\tilde U_i\}_{i \in I}\) a fundamental system of symmetric entourages in \(Y\). It is clear that the set of indices in both cases may be taken to be one and the same. With each entourage \(V\) we shall associate a sequence of entourages \(V^{(1)}, \ldots, V^{(n)}, \ldots\), satisfying the conditions
\[ V^{(1)}V^{(1)} \subset V;\qquad V^{(n+1)}V^{(n+1)} \subset V^{(n)} \quad (n=1,2,\ldots). \]
Following (³), a multivalued mapping \(F\) of the space \(X\) into the space \(Y\) will be called uniformly almost open (respectively, uniformly open) if for every entourage \(U\) in \(X\) there exists an entourage \(U^{F}\) in \(Y\) such that, for every point \(x_0 \in X\), the inclusion
\[ \overline{FU}(x_0) \supset U^{F}F(x_0) \]
holds (respectively,
\[ FU(x_0) \supset U^{F}F(x_0)). \]

The Hausdorff uniformity on the set \(\mathfrak F(X)\) of all closed subsets of the space \(X\) is the uniformity defined by the base of entourages
\[ \{(A,B): A \subset U(B),\ B \subset U(A)\}, \]
where \(U\) runs through any fundamental system of entourages of the uniformity given on \(X\). If \(\mathfrak F(X)\), endowed with the Hausdorff uniformity, is complete, then the space \(X\) is called ultracomplete.

Lemma*. Let \(F\) be a uniformly almost open multivalued mapping with closed graph from an ultracomplete space \(X\) into an arbitrary uniform space \(Y\). Then, for any entourages \(U\) and \(V\) in \(X\) and any point \(x_0 \in X\), the inclusion
\[ FVU(x_0) \supset \overline{FU}(x_0) \]
is valid.

Proof. Obviously, one may assume that
\[ U_i^{(n)} \subset V^{(n+1)} \]
for all \(i\) and all \(n\).

Let
\[ \bar y \in \overline{FU}(x_0). \]
We shall construct a fundamental (in the sense of the Hausdorff uniformity) generalized sequence of sets

* In the special case when \(X\) is a complete pseudometric space, this assertion was proved by J. Kelley ((⁴), pp. 202–203).

\(C_\mu \subset X\), such that: 1) all \(C_\mu\) lie in \(V^{(1)}U(x_0)\), so that \(C=\lim_\mu \overline{C}_\mu\) lies in \(VU(x_0)\); 2) \(\bar y\in F(\bar x)\) for any point \(\bar x\in C\); 3) the index \(\mu\) runs through all possible finite collections of elements of the set \(I\) (ordered by inclusion).

First we construct an auxiliary sequence of sets \(B_\mu\) satisfying the conditions: a) all \(B_\mu\) lie in \(V^{(1)}U(x_0)\); b) if \(\mu \supset \nu\), then

\[ B_\nu \subset \bigcap_{i\in\nu} U_i^{(|\nu|)}(B_\mu), \]

where \(|\nu|\) is the number of elements of the set \(\nu\); c) for any point \(x\in B_\mu\) the relation

\[ \bigcap_{i\in\mu} \widetilde U_i(\bar y)\cap F(x)\ne \varnothing \]

holds.

If such sets \(B_\mu\) have been constructed, then it suffices to put

\[ C_\mu=\bigcup_{\nu\supset\mu} B_\nu . \]

Indeed, let us verify the fundamentality of the system \(\{C_\mu\}\). If \(\mu\supset\nu\), then \(C_\mu\subset C_\nu\). Therefore, for each neighborhood \(W\) it is necessary to find a \(\mu_0\) such that from \(\mu\supset\mu_0\) it follows that \(W(C_\mu)\supset C_{\mu_0}\). As such a \(\mu_0\) take a set consisting of one index \(i\) satisfying \(U_i\subset W\). Let \(x\in C_i\); then \(x\in B_\nu\), where \(i\in\nu\), and let \(\xi=\mu\cup\nu\). Then \(B_\xi\subset C_\mu\), and

\[ x\in B_\nu \subset \bigcap_{j\in\nu} U_j^{(|\nu|)}(B_\xi)\subset U_i(B_\xi)\subset W(B_\xi), \]

as was required. Thus, the sequence \(C_\mu\) is fundamental and satisfies conditions 1) and 3). Let us verify condition 2).

If \(\bar x\in C\), then every neighborhood of the point \(\bar x\) contains points from \(C_\mu\) for all sufficiently large \(\mu\), and hence also points from \(B_\nu\) for \(\nu\) belonging to some cofinal subsequence of the set of indices. From this and from property c) of the sets \(B_\nu\) it follows that for given neighborhoods \(U\) and \(\widetilde U\) of the points \(\bar x\) and \(\bar y\), respectively, there exist points \(x\in U(\bar x)\) and \(y\in \widetilde U(\bar y)\) for which \(y\in F(x)\), i.e. \((x,y)\in F\). From the closedness of the graph it follows that \(\bar y\in F(\bar x)\).

It remains to construct the sets \(B_\mu\). We shall construct them by induction on \(|\mu|\), the number of elements in \(\mu\).

\[ B_i=\{x:\ x\in U(x_0),\ \widetilde U_i(\bar y)\cap U_i^F(\bar y)\cap F(x)\ne \varnothing\}; \]

\[ B_\mu=\{x:\ x\in \bigcup_{\substack{\nu\subset\mu\\ \nu\ne\mu}}\bigcap_{i\in\nu} U_i^{(|\nu|)}(B_\nu),\ \left(\bigcap_{i\in\mu}\widetilde U_i(\bar y)\right)\cap \left(\bigcap_{i\in\mu} U_i^{(|\mu|)}\right)^F(\bar y)\cap F(x)\ne \varnothing\}. \]

Since

\[ \bar y\in \overline{FU(x_0)}, \]

the intersection of any neighborhood of the point \(\bar y\) with the set \(FU(x_0)\) is nonempty. Hence the nonemptiness of the sets \(B_i\) follows. At the same time it is easy to see that \(\bar y\in \overline{F(B_i)}\), and, consequently, the same reasoning proves the nonemptiness of the sets \(B_\mu\). Let us prove that requirements a), b), c) are satisfied.

For all \(B_i\) the inclusion \(B_i\subset U(x_0)\) is satisfied. Suppose that for all \(\nu\) for which \(|\nu|<k\), the inclusion

\[ B_\nu\subset V^{(|\nu|)}V^{(|\nu|-1)}\ldots V^{(2)}U(x_0) \]

is satisfied, and let \(|\mu|=k\). Then

\[ B_\mu\subset \bigcup_{\substack{\nu\subset\mu\\ \nu\ne\mu}} \bigcap_{i\in\nu} U_i^{(|\nu|)}(B_\nu) \subset \bigcup_{\substack{\nu\subset\mu\\ \nu\ne\mu}} V^{(|\nu|+1)}(B_\nu) \subset \]

\[ \subset \bigcup_{\substack{\nu\subset\mu\\ \nu\ne\mu}} V^{(|\nu|+1)}V^{(|\nu|)}\ldots V^{(2)}U(x_0) = V^{(|\mu|)}\ldots V^{(2)}U(x_0). \]

It follows that all \(B_\mu\) are contained in \(V^{(1)}U(x_0)\), i.e. assertion a).

Now let \(x\in B_\nu\) and \(\mu\supset\nu\). Then from the relation

\[ \left(\bigcap_{i\in\nu} U_i^{(|\nu|)}\right)^F(\bar y)\cap F(x)\ne \varnothing, \]

i.e.

\[ \bar y\in \left(\bigcap_{i\in\nu} U_i^{(|\nu|)}\right)^F F(x) \subset \overline{F\left(\bigcap_{i\in\nu} U_i^{(|\nu|)}\right)(x)} \]

there follows the exist—

of the existence of a point \(x' \in \bigcap_{i\in\nu} U_i^{(|\nu|)}(x)\) such that

\[ \left(\bigcap_{i\in\mu} \overline{U_i}(\bar y)\right)\cap \left(\bigcap_{i\in\mu} U_i^{(|\mu|)}\right)^F(\bar y)\cap F(x')\ne \varnothing, \]

and this means that \(x'\in B_\mu\) (recall that \(x\in B_\nu\)). Using the symmetry of the neighborhoods, we obtain \(x\in \bigcap_{i\in\nu} U_i^{(|\nu|)}(x')\), i.e. \(B_\nu\subset \bigcap_{i\in\nu} U_i^{(|\nu|)}(B_\mu)\). Thus condition b) is fulfilled. Property c) follows from the definition of the sets \(B_\mu\).

The lemma is proved. It immediately implies the following

Theorem. If the space \(X\) is ultracomplete, then every uniformly almost open multivalued mapping with closed graph from the space \(X\) into an arbitrary uniform space \(Y\) is uniformly open.

Tashkent State University
named after V. I. Lenin

Received
10 III 1964

CITED LITERATURE

¹ V. Pták, Bull. Soc. math. France, 86, 41 (1958); Collected Transl., Mathematics, 4, 6 (1960). ² J. L. Kelley, Michigan Math. J., 5, 235 (1958); Collected Transl., Mathematics, 4, 6 (1960). ³ V. L. Levin, D. A. Raikov, DAN, 150, No. 5 (1963). ⁴ J. L. Kelley, General Topology, N. Y., 1955.