MATHEMATICS

V. L. LEVIN, D. A. RAIKOV

CLOSED-GRAPH THEOREMS FOR UNIFORM SPACES

(Presented by Academician P. S. Novikov on January 11, 1963)

V. Pták \(\left(^{1,2}\right)\), analyzing the well-known Banach theorem on open mappings \(\left(^{3}\right)\), introduced a class of locally convex spaces, called by him \(B\)-complete and characterized by the fact that every “almost open” continuous linear mapping of them onto any locally convex space is open. A. and W. Robertson \(\left(^{4}\right)\) extended to \(B\)-complete spaces also Banach’s closed-graph theorem \(\left(^{3}\right)\), and then Pták \(\left(^{5}\right)\) embraced both properties by a single theorem, which is essentially a closed-graph theorem for multivalued linear mappings of \(B\)-complete spaces. Pták used methods of the duality theory of locally convex spaces. But J. Kelley \(\left(^{6}\right)\) showed that \(B\)-completeness of a locally convex space is equivalent to a certain property of weakened completeness of the space of its closed subsets, endowed with the “Hausdorff uniformity,” and, using this, obtained a significant part of the results of Pták and the Robertsons without the aid of duality theory. This made it possible, by a small modification of Kelley’s arguments, to extend the theory of \(B\)-completeness also to non-locally convex topological linear spaces \(\left(^{7}\right)\).

The purpose of the present note is to transfer the theory of \(B\)-completeness to uniform spaces.

1. Let \(E\) and \(F\) be nonempty sets. A multivalued mapping (in what follows, more often simply a mapping) from \(E\) to \(F\) is a mapping \(f\) assigning to each element \(x \in E\) a set \(fx \subset F\). The domain of definition of \(f\) is the set \(D_f\) of those \(x \in E\) for which the image \(fx\) is nonempty. \(f\) is called single-valued if \(fx\) is a singleton for every \(x \in D_f\). \(f\) is called a mapping of \(E\) into \(F\) (and not “from \(E\) into \(F\)”) if \(D_f = E\).

Let \(f\) be a mapping from \(E\) to \(F\). The mapping \(f^{-1}\) from \(F\) to \(E\), inverse to \(f\), is defined by the condition that the relations \(x \in f^{-1}y\) and \(y \in fx\) are equivalent. The graph of \(f\) is the set \(\{(x,y) \in E \times F: y \in fx\}\). By \(\tilde f\) is meant \(f \times f\), i.e., the mapping from \(E \times E\) to \(F \times F\) given by the formula

\[ \tilde f(x_1,x_2)=\{(y_1,y_2)\in F_1\times F_2:\ y_1\in fx_1,\ y_2\in fx_2\}. \]

\(R \subset E \times E\) (as well as a mapping from \(E\) to \(E\) with graph \(R\)) is called a partial equivalence if \(R^2 = R^{-1} = R\), or, equivalently, \(R^{-1}R = R\).

1. The uniform space obtained by endowing the set \(E\) with a uniformity \(u\) (which we shall identify with the collection of its entourages) is denoted by \((E,u)\); \((E,u)\) is not assumed to be separated.

We shall say that a mapping \(f\) from a uniform space \((E,u)\) to a uniform space \((F,v)\) is uniformly continuous if for every \(V \in v\) there exists a \(U \in u\) such that \(Ux \subset f^{-1}Vy\) for all \(y \in F\) and all \(x \in f^{-1}y\); uniformly almost continuous if for every \(V \in v\) there exists a \(U \in u\) such that \(Ux \subset \overline{f^{-1}Vy}\) for all \(y \in F\) and all \(x \in f^{-1}y\) (where the bar denotes closure); uniformly open if \(f^{-1}\) is uniformly continuous; uniformly almost open if \(f^{-1}\) is uniformly almost continuous; uniformly

homeomorphic if \(f\) is uniformly continuous and uniformly open; it has a closed graph if the graph of \(f\) is closed in the uniform space \((E,u)\times(F,v)\).

1. Let \(f\) be a mapping from a uniform space \((E,u)\) into a set \(F\). We shall say that \(f\) is regular if \(\tilde f(u)\) is a uniformity on \(f(E)\), and \(f\) is a uniformly open mapping from \((E,u)\) onto \((f(E),\tilde f(u))\). We shall say that the mapping \(f\) from \((E,u)\) into \((F,v)\) is biregular if \(f\) is a regular mapping from \((E,u)\) into \(F\), and \(f^{-1}\) is a mapping from \((F,v)\) into \(E\).

Examples. a) A one-to-one mapping from \((E,u)\) into \(F\) (respectively from \((E,u)\) into \((F,v)\)) is regular (respectively biregular).
b) A mapping inverse to a single-valued mapping from \(F\) into \((E,u)\) is regular.
c) A single-valued uniformly homeomorphic mapping from \((E,u)\) into \((F,v)\) is biregular.
d) Projections of a product of uniform spaces onto these spaces are biregular.
e) An algebraic homomorphism from a topological group \(G\) into a group \(H\) (respectively into a topological group \(H\)) is regular (respectively biregular).
f) The canonical mapping of a topological group \(G\), endowed with the right uniformity, onto the homogeneous space \(G/H\) of left cosets modulo the subgroup \(H\) is biregular.

Theorem 1. If \(f\) is a regular mapping with closed graph from \((E,u)\) into \((F,v)\), then the images \(fx_1, fx_2\) of points \(x_1,x_2\in E\), as well as the inverse images \(f^{-1}y_1, f^{-1}y_2\) of points \(y_1,y_2\in F\), either coincide or do not intersect.

1. Let \((E,u)\) be a uniform space. Denote by \(\mathscr E_u\) the uniform space whose elements are all nonempty subsets of the product \(E\times E\), and whose uniformity is given by the base of entourages \(\{(M,N): M\subset UN,\; N\subset UM\}\), where \(U\) ranges over \(u\).

A \(B\)-net for \((E,u)\) will mean a net (generalized sequence) \(\{W_\alpha\}_{\alpha\in A}\) in \(\mathscr E_u\) satisfying the following conditions: 1) \(\{W_\alpha\}\) is a fundamental system of symmetric entourages of some uniformity on some set \(E_0\subset E\); 2) the partial equivalence
\[ R=\bigcap_{\alpha\in A} W_\alpha \]
is regular; 3)
\[ \bigcap_{\alpha\in A,\; U\in u} UW_\alpha = R; \]
4) \(\{W_\alpha\}_{\alpha\in A}\) is a Cauchy net in \(\mathscr E_u\) (i.e., for every \(U\in u\) there exists \(\alpha_U\in A\) such that
\[ W_{\alpha'}\subset UW_\alpha \]
for all \(\alpha,\alpha'\geq \alpha_U\)).

We shall call \((E,u)\) \(B\)-complete if every \(B\)-net \(\{W_\alpha\}_{\alpha\in A}\) in \(\mathscr E_u\) converges (then it converges to \(R\), i.e., for every \(U\in u\) there exists \(\alpha_U\in A\) such that
\[ W_\alpha\subset UR \]
for all \(\alpha\geq \alpha_U\)).

Theorem 2. The following assertions about a uniform space \((E,u)\) are equivalent:

1) \((E,u)\) is \(B\)-complete;

2) every uniformly nearly continuous biregular mapping with closed graph of any uniform space \((F,v)\) into \((E,u)\) is uniformly continuous;

3) every uniformly nearly open biregular mapping with closed graph from \((E,u)\) onto any uniform space \((F,v)\) is uniformly open;

4) every single-valued uniformly nearly open regular mapping with closed graph from \((E,u)\) onto any uniform space \((F,v)\) is uniformly open.

1. A \(B'\)-net for \((E,u)\) will mean any \(B\)-net for which \(E_0=E\). We shall call the space \((E,u)\) \(B'\)-complete if every \(B'\)-net in \(\mathscr E_u\) converges. Obviously, every \(B\)-complete space is \(B'\)-complete.

Theorem 2′. The following assertions about a uniform space \((E,u)\) are equivalent:

1′) \((E,u)\) is \(B'\)-complete;

2′) every uniformly nearly continuous biregular mapping-

with closed graph from any uniform space \((F, v)\) onto \((E, u)\) is uniformly continuous;

3′) every uniformly nearly open biregular mapping with closed graph from the space \((E, u)\) onto any uniform space \((F, v)\) is uniformly open;

4′) every single-valued uniformly nearly open regular mapping with closed graph from the space \((E, u)\) onto any uniform space \((F, v)\) is uniformly open;

5′) every single-valued uniformly nearly continuous regular mapping with closed graph from any uniform space \((F, v)\) onto the quotient space \((E/R, u/R)\) of the space \((E, u)\) by a regular equivalence \(R\) is uniformly continuous.

1. Let \(\mathfrak p(E)\) be the set of all nonempty subsets of the uniform space \((E, u)\). The Hausdorff uniformity in \(\mathfrak p(E)\) is the uniformity \(\mathfrak p(u)\) defined by the base of entourages

\[ \{(A, B): A \subset U(B),\ B \subset U(A)\}, \]

where \(U\) runs through \(u\). \((E, u)\) is called ultracomplete if \((\mathfrak p(E), \mathfrak p(u))\) is complete.

\((E, u)\) is ultracomplete if and only if \(\mathcal E_u\) is complete. Thus every ultracomplete space is \(B\)-complete, and hence possesses the “closed graph properties” of Theorems 2—2′.

Theorem 3. The quotient space \((E/R; u/R)\) of an ultracomplete (respectively \(B\)-complete, \(B'\)-complete) space \((E, u)\) by a regular equivalence \(R\) is ultracomplete (respectively \(B\)-complete, \(B'\)-complete).

As is known, complete metrizable uniform spaces are ultracomplete. Another example of ultracomplete spaces is provided by uniformly locally bicompact spaces. Necessary and sufficient conditions for ultracompleteness of a uniform space were recently found by Isbell \((^8)\).

Received
11 I 1963

CITED LITERATURE

\(^1\) V. Pták, Czechoslovak Math. J., 3(78), 301 (1953). \(^2\) V. Pták, Bull. Soc. math. France, 86, 41 (1958); Russian transl. in Matematika, 4, 6 (1960). \(^3\) S. Banach, Course of Functional Analysis, Kiev, 1948. \(^4\) A. Robertson, W. Robertson, Proc. Glasgow Math. Assoc., 3, 9 (1956); Russian transl. in Matematika, 4, 6 (1960). \(^5\) V. Pták, Czechoslovak Math. J., 9, 523 (84) (1959); Russian transl. in Matematika, 4, 6 (1960). \(^6\) J. L. Kelley, Michigan Math. J., 5, 235 (1958); Russian transl. in Matematika, 4, 6 (1960). \(^7\) D. A. Raikov, Proceedings of the Fourth All-Union Mathematical Congress (in press). \(^8\) J. R. Isbell, Pacific J. Math., 12, 1 (1962).