UDC 513.83

MATHEMATICS

A. D. GORBUNOV

ON THE CONTINUITY OF MAPPINGS AND TRANSLATIONS

(Presented by Academician Yu. N. Rabotnov on 28 VI 1966)

1. Let a topological space \(R\) be given, in which a complete system of neighborhoods is defined (1). We shall say that a complete system of neighborhoods is defined for every set \(M \subset R\), if for \(M\) a system \(\Sigma(M)\) of sets open in \(R\) is given, possessing the following properties: a) for every set \(U \in \Sigma(M)\) the inclusion \(M \subset U\) holds; b) for every pair of sets \(U\) and \(V\) from \(\Sigma(M)\) there exists a set \(W \in \Sigma(M)\) such that \(W \subset U \cap V\); c) for every set \(U \in \Sigma(M)\) and every set \(M' \subset \overline{M}\) there exists a set \(V \in \Sigma(M')\) such that \(V \subset U\); d) if \(M' \subset M\), then for every set \(U \in \Sigma(M)\) there is a set \(U' \in \Sigma(M')\) such that \(U' \subset U\); e) if a sequence of points convergent in \(R\) is almost entirely contained in an arbitrary set \(U\) of the system \(\Sigma(M)\), then its limit is a point of contact of the set \(M\); f) if the set \(M\) consists of one point, then \(\Sigma(M)\) coincides with the complete system of neighborhoods of this point. The totality of all neighborhoods of all sets in \(R\) will be called the complete system of generalized neighborhoods of the space \(R\). It is assumed that all topological spaces occurring in the article are, once and for all, equipped with neighborhoods and generalized neighborhoods; moreover, in metric spaces, as neighborhoods there are taken, as usual, all possible spheres of points, and as generalized neighborhoods—all possible “spheres” of sets.

2. A point sequence \(x_k \in R,\ k = 1, 2, \ldots,\) will be called bounded in \(R\) if there exists a neighborhood of the space \(R\) containing almost all elements of this sequence. We shall assume that sequences not bounded in \(R\) can be distinguished from one another by the character of unboundedness or by order of growth.

We shall distinguish point sequences in \(R\) by types. Every point sequence convergent in \(R\) will be assigned to the zeroth type, a bounded one—to the second type, an unbounded sequence with growth not exceeding a certain order—to the fourth type, and, finally, an arbitrary point sequence in \(R\)—to the sixth type.

1. Let a sequence of sets \(M_k,\ k = 1, 2, \ldots\), be given in \(R\). A sequence of sets \(M_k^*,\ k = 1, 2, \ldots,\) will be called a daughter sequence for the sequence \(M_k\), if for every \(k\) the inclusion \(M_k^* \subset M_k\) is satisfied.

We shall say that a sequence of sets \(M_k,\ k = 1, 2, \ldots,\) converges in \(R\) of the \(i\)-th type to the set \(M \subset R\) \((i = 0, 2, 4, 6)\), if for every neighborhood \(V\) of the set \(M\) and every sequence of points \(x_k,\ k = 1, 2, \ldots,\) of the \(i\)-th type and daughter for the sequence \(M_k\), under the condition that the set of such sequences is nonempty, there is a number \(k_0\) such that from the inequality \(k \ge k_0\) there follows the inclusion \(x_k \in V\).

We shall say that a sequence of sets \(M_k,\ k = 1, 2, \ldots,\) converges in \(R\) of the \((i+1)\)-st type to the set \(M \subset R\) \((i = 0, 2, 4, 6)\), if it converges in \(R\) to the set \(M\) of the \(i\)-th type and for every point \(x \in M\)

there is a sequence of points \(x_k,\ k=1,2,\ldots,\) subordinate to the sequence \(M_k\), for which the point \(x\) is a limit point.

If a sequence of sets \(M_k,\ k=1,2,\ldots,\) converges in \(R\) of the seventh type to a set \(M\), then it converges in \(R\) of the seventh type also to the set \(\overline M\), the latter being uniquely determined and naturally called the limit of the seventh type of the sequence \(M_k\).

Let \(\sigma_i\) denote the type of convergence in \(R\) of a sequence of sets to some set \((i=0,1,2,3,4,5,6,7)\). It is easy to see that the following implications hold:

\[ \sigma_6 \to \sigma_0,\qquad \sigma_6 \to \sigma_2,\qquad \sigma_6 \to \sigma_4; \]

\[ \sigma_7 \to \sigma_1,\qquad \sigma_7 \to \sigma_3,\qquad \sigma_7 \to \sigma_5; \]

\[ \sigma_1 \to \sigma_0,\qquad \sigma_3 \to \sigma_2,\qquad \sigma_5 \to \sigma_4,\qquad \sigma_7 \to \sigma_6. \]

4. Let two spaces \(R_1\) and \(R_2\) be given. We shall say that a translation \(B\) from \(R_1\) into \(R_2\) is defined if a rule is specified by which a set \(P_B \subset R_1\) is determined, and to each element \(x\) of \(P_B\) there is assigned a definite set \(\eta\) of elements of \(R_2\). In this case we shall write \(\eta = Bx\). As \(x\) ranges over \(P_B\), \(\eta \subset R_2\). The set \(P_B\) will be called the domain of definition of the translation \(B\). The union of all sets \(\eta = Bx\), as \(x\) ranges over \(P_B\), will be denoted by \(Q_B\) and called the range* of the translation \(B\). It is clear that \(B\) is a translation of \(P_B\) onto \(Q_B\): \(\eta = Bx\), \(x\) ranges over \(P_B\), and \(Q_B\) is the union of the sets \(\eta = Bx\), \(x\) ranges over \(P_B\).

If the set \(\eta = Bx\) consists of one point, then \(x\) is called a point of single-valuedness of the translation \(B\). If all points of \(P_B\) are points of single-valuedness for \(B\), then \(B\) is a mapping of \(P_B\) onto \(Q_B\).

If a translation \(B\) of the space \(P_B\) onto \(Q_B\) is defined, then thereby also defined is the translation which assigns to each point \(y \in Q_B\) the set \(\xi\) of all elements \(x\) of \(P_B\) for which \(y \in Bx\). This translation is called the inverse of \(B\) and is denoted by the symbol \(B^{-1}\). If the inverse of the translation \(B\) is a mapping, then the translation \(B\) is called one-to-one; otherwise, many-to-one.

Assign to each point \(x \in P_B\) a definite subset \(B^*x\) of the set \(Bx\). As a result there arises a translation \(B^*\) of the space \(P_B\) into \(Q_B\), which we shall call subordinate to the translation \(B\).

We shall say that a family of mappings \(B_\alpha\) exhausts the translation \(B\) if, for every \(\alpha\), the mapping \(B_\alpha\) is subordinate to \(B\), and for every \(x \in P_B\) the equality
\[ Bx = \bigcup_\alpha B_\alpha x \]
holds.

5. Let \(R_1\) and \(R_2\) be topological spaces. A translation \(B\) from \(R_1\) into \(R_2\) will be called complete if, for every \(x \in P_B\), the set \(Bx\) is closed. A translation \(B\) will be called regular if, for every \(x \in P_B\), the set \(Bx\) is bounded. A translation \(B\) from \(R_1\) into \(R_2\) will be called continuous at the point \(x=x_0\) of the \(i\)-th type \((x_0 \in P_B,\ i=0,1,2,3,4,5,6,7)\) if, for every sequence of points \(x_k \in P_B,\ k=1,2,\ldots,\) converging in \(R_1\) to the point \(x_0\), the sequence \(Bx_k,\ k=1,2,\ldots,\) converges in \(R_2\) of the \(i\)-th type to \(Bx_0\). A translation \(B\) from \(R_1\) into \(R_2\) will be called continuous of the \(i\)-th type if at each point of the set \(P_B\) it is continuous of the \(i\)-th type.

Theorem 1. If a translation \(B\) from \(R_1\) into \(R_2\) is continuous of the \(i\)-th type \((i=0,1,2,3,4,5,6,7)\), then every restriction of the translation \(B\) is continuous of the \(i\)-th type.

Theorem 2. If a mapping \(B\) of the space \(R_1\) into \(R_2\) is continuous and, for every sequence of points \(y_k \in Q_B,\ k=1,2,\ldots,\) converging in \(R_2\) to a point \(y_0 \in Q_B\), for the sequence of sets \(B^{-1}y_k,\ k=1,2,\ldots,\)

* In this section, by a space is meant a nonempty fixed set of some elements (?).

there exists a daughter sequence of points of zero type, then the translation \(B^{-1}\) is discontinuous of zero type at the point \(y_0\).

Theorem 3. If the mapping \(B\) of the space \(R_1\) into \(R_2\) is discontinuous and open, then the translation \(B^{-1}\) is discontinuous of the first type.

Theorem 4. If the mapping \(B\) of the space \(R_1\) into \(R_2\) is open, then the image of every closed set is a closed set.

1. Let \(R\) be a metric space with metric \(\rho\). Consider the functional

\[ \Psi_x y = \rho(x,y),\quad y \text{ ranges over } \overline{Q}, \]

where \(Q\) is a nonempty subset of \(R\) and \(x \in R\).

Theorem 5. If the set \(Q\) is dense in \(R\), or if every sphere of the space \(R\) is compact, then the functional \(\Psi_x\) has a minimum; moreover, the set \(M_x\) of elements minimizing this functional is a full regular translation of the space \(R\) into \(\overline{Q}\), discontinuous of the sixth type.

1. Let \(R_1\) and \(R_2\) be metric spaces with metrics \(\rho_1\) and \(\rho_2\), respectively. A family of mappings \(B_\alpha\) from \(R_1\) into \(R_2\), defined on one and the same set \(P \subset R_1\), will be called uniformly discontinuous at a point \(x_0 \subset P\) if, for every \(\varepsilon > 0\), there exists a neighborhood \(V_{x_0} \subset R_1\) of the point \(x_0\) such that the inclusion

\[ B_\alpha(V_{x_0}\cap P)\subset s(B_\alpha x_0,\varepsilon)\cap B_\alpha P. \]

holds.

Theorem 6. Let a translation \(B\) from \(R_1\) into \(R_2\) be given, and let a family of mappings \(B_\alpha\) exhausting it be defined. Then, if the family \(B_\alpha\) is uniformly discontinuous at each point of the set \(P_B\), the translation \(B\) is discontinuous of the seventh type.

Denote by \(R_3\) the topological product of the spaces \(R_1\) and \(R_2\), with metric \(\rho_3=\max(\rho_1,\rho_2)\). If \(K_1\) and \(K_2\) are compact-in-themselves subsets of the spaces \(R_1\) and \(R_2\), respectively, then \(K_3\), equal to the topological product of \(K_1\) by \(K_2\), will be a compact-in-itself subset of the space \(R_3\), and conversely.

Theorem 7. Let a real continuous functional \(F(x,y)\) be given on a compact-in-itself set \(K_3\) of the space \(R_3\), where \(x\) ranges over \(K_1\) and \(y\) ranges over \(K_2\). We shall regard \(F(x,y)\) as a functional only of \(y\) for fixed \(x\): \(F_x y = F(x,y)\), \(y\) ranges over \(K_2\). Then the set \(N_x\) of elements minimizing the functional \(F_x y\), where \(y\) ranges over \(K_2\), is a full, regular translation of \(K_1\) into \(K_2\), discontinuous of the sixth type.

1. Theorem 8. Let a translation \(B\) be given from the topological space \(R_1\) onto the topological space \(R_3\). Suppose, moreover, that a translation \(T\) of the subspace \(P_B\) onto the space \(R_2\) and a translation \(S\) of the space \(R_2\) onto the space \(R_3\) are given such that \(B=ST\).

A. Then, if the space \(R_2\) is topological, the translation \(S\) is discontinuous of the zero (first) type, and the translation \(T\) is full and discontinuous of the zero (first) type, then the translation \(B\) is discontinuous of the zero (first) type.

B. Then, if the space \(R_2\) is metric and each of its closed spheres is compact, the translation \(T\) is full and regular, and the translations \(S\) and \(T\) are discontinuous of the sixth type, then the translation \(B\) is discontinuous of the sixth type.

Remark. Theorem 8B is also true in the case when the set \(Q_B\) is a proper part of the space \(R_3\).

Moscow State University
named after M. V. Lomonosov

Received
17 VI 1966

CITED LITERATURE

1. L. S. Pontryagin, Continuous Groups, Moscow, 1954.
2. L. Collatz, Funktionalanalysis und numerische Matematik, Berlin, 1964.