UDC 519.50/54

MATHEMATICS

V. V. MASLOV

ON THE THEORY OF SEMISYNTOPOGENIC SPACES

(Presented by Academician P. S. Novikov on 24 XI 1967)

For the definitions see (¹).

§ 1. A semisyntopogenic space is a pair \([E; S]\), where \(E\) is some set, and \(S\) is a nonempty set of semitopogenic orders on \(E\) satisfying two axioms:

\((S_1)\) for any two orders \(<_{1}, <_{2} \in S\) there is a stronger order \(<_{3} \in S\).

\((S_2)\) if \(< \in S\), then there exists an order \(<_{1} \in S\) such that \(<_{1}^{2} \supset <\).

If every order \(< \in S\) is topogenic, then the space \([E; S]\) is called syntopogenic. Such spaces were studied by A. Császár (¹). It turns out that the concept of a semisyntopogenic space is essentially broader than the concept of a syntopogenic space (for the corresponding example, though in another connection, see § 3).

Having a semisyntopogenic space \([E; S]\), one may speak of \(S\)-open sets (\(A \subset E\) is \(S\)-open if \(A < A\) for some \(< \in S\)), \(S\)-closed sets (\(A \subset E\) is \(S\)-closed if \(S[A] \subset A\); here \(S[A]\) denotes the set of limit points of the set \(A\) with respect to \(S\): \((x \in S[A]) \Longleftrightarrow (\forall B)(B \in V(x) \Rightarrow B \cap A \ne \varnothing)\); we assume here that the set \(V(x)\) of neighborhoods of the point \(x\) with respect to the structure \(S\) forms a filter).

Having obtained the notions of an \(S\)-open and an \(S\)-closed set, one may define, for example, the notion of connectedness of a semisyntopogenic space \([E; S]\). It turns out that a generalization of the classical concept of a connected topological space is possible in two nonequivalent ways; namely, the definition of a connected space by means of \(S\)-open sets is strictly included in the definition of a connected space by means of \(S\)-closed sets. It is interesting to note the following

Proposition. If a semisyntopogenic space \([E; S]\), in which the \(S\)-open and \(S\)-closed sets are mutually complementary, is \(S\)-disconnected, then there exists an \((S; S')\)-continuous mapping of this space onto a discrete space \([E'; S']\).

A corresponding sufficient condition could not be formulated so simply.

§ 2. Let \(X\) be some set, and let \([E; S]\) be a semisyntopogenic space; by \(F(X; E)\) denote the set of mappings from \(X\) into \(E\). Take some order \(<\) and define the relation \(<'\) between subsets of the set \(F(X; E)\) as follows: \(A <'' B\) means that the set \(A\) consists of such and only such mappings \(u : X \to E\) for which \(u(X) \subset A_{1}\), and the set \(B\) consists of such and only such mappings \(v : X \to E\) for which \(v(X) \subset B_{1}\), with \(A_{1} < B_{1}\),

\[ (A_{2} <' B_{2}) \Longleftrightarrow (\exists A_{1})(\exists B_{1})(A_{2} \subset A_{1} <'' B_{1} \subset B_{2}). \]

It turns out that \(<'\) is a semitopogenic order, and the set

\[ S'=\{<': < \in S\} \]

is a semisyntopogenic structure on \(F(X; E)\). Let

\[ \mathfrak{X}=(X_{\alpha})_{\alpha \in A} \]

be a nonempty family of nonempty subsets of the set \(X\); by \(\varphi_{\alpha}\) denote the mapping \(u \to u|X_{\alpha}\) (from \(F(X; E)\) to \(F(X; E_{\alpha})\)); struct-

the structure \(S'_{\mathfrak X}\) on \(F(X;E)\) is defined as the supremum of the family of structures \(\varphi_\alpha^{-1}(S'_\alpha)\), \(\alpha\in A\). It is interesting to note the construction of the structure \(S'_{\mathfrak X}\).

Let
\[ \varphi:\ u\to (u|X_\alpha)_{\alpha\in A} \]
be the canonical mapping of the set \(F(X;E)\) into the set
\[ \prod_{\alpha\in A}F(X_\alpha;E); \]
it turns out that the structure \(S'_{\mathfrak X}\) is majorized by the structure
\[ \varphi^{-1}\left(\prod_{\alpha\in A}S'_\alpha\right), \]
and if \(S'_{\mathfrak X}\) is syntopogenous, then these two structures are isomorphic.

Theorem 1. Let \(X\) be some set, \([Y;S]\) a semisyntopogenous space, and \(F\) a subset of the set \(F(X;Y)\) consisting of constant mappings. The spaces \([Y;S]\) and \([F;S'_{\mathfrak X}|F]\) are isomorphic.

Theorem 2. If the space \([F(X;Y);S'_{\mathfrak X}]\) is \(T_2\)-separated, then the space \([Y;S]\) is also \(T_2\)-separated.

Theorem 3. Suppose that a nonempty family \(\mathfrak X=(X_\alpha)_{\alpha\in A}\) of nonempty parts of a set \(X\) contains, together with each set \(X_\alpha\), all subsets of this set. If the family \(\mathfrak X=(X_\alpha)_{\alpha\in A}\) covers the set \(X\), and the semisyntopogenous space \([Y;S]\) is \(T_2\)-separated, then the space \([F(X;Y);S'_{\mathfrak X}]\) is \(T_2\)-separated.

Theorem 4. Let \(X\) and \(Y\) be some nonempty sets, \([Y_\mu;S_\mu]\), \(\mu\in M\), a nonempty family of semisyntopogenous spaces, \((X_\lambda)_{\lambda\in L}\) a family of sets distinct from the empty set, and, for each index \(\lambda\in L\),
\[ \mathfrak X_\lambda=(X_{\alpha_\lambda})_{\alpha_\lambda\in A_\lambda} \]
a nonempty family of nonempty parts of the set \(X_\lambda\). Let, further,
\[ \psi_\lambda:X_\lambda\to X,\qquad f_\mu:Y\to Y_\mu \]
for \(\lambda\in L\) and \(\mu\in M\). If the set \(Y\) is endowed with the structure
\[ S=\bigvee_{\mu\in M} f_\mu^{-1}(S_\mu) \]
and one puts
\[ \mathfrak X=(X_{\alpha_\lambda}),\qquad \alpha^\lambda\in A_\lambda, \]
for all \(\lambda\in L\), then the mapping
\[ g_{\lambda\mu}:\ u\to f_\mu\circ u\circ\psi_\lambda \]
(where \(u\in (X;Y)\)) is \((S'_{\mathfrak X};(S_\mu)'_{\mathfrak X_\lambda})\)-continuous for all \((\lambda;\mu)\in L\times M\).

§ 3. The study of semisyntopogenous structures together with algebraic ones defined on one and the same set and naturally coordinated with one another is of known interest.

We call a semisyntopogenous space \([E;S]\) a semisyntopogenous group if \(E\) is a group and the mapping
\[ (x;y)\to x\cdot y^{-1} \]
is \(((S\times S)^p;S)\)-continuous.

We shall note only the following fact. From a given structure \(S\) on \(E\) one can define structures \(S_\pi\) and \(S_\lambda\), which in the case when \([E;S]\) is a topological group coincide respectively with the right and left uniform structures on \(E\).

Theorem 5. If the group \(E\) is commutative, then \([E;S_\pi]\) is a semisyntopogenous group.

Analogously one can define the notions of semisyntopogenous fields and vector spaces. Here are some theorems concerning the latter notion.

Theorem 6. For \((S_1;S_2)\)-continuity of a linear mapping \(f\) from a semisyntopogenous vector space \([E_1;S_1]\) into a semisyntopogenous vector space \([E_2;S_2]\), it is sufficient that the following conditions be fulfilled: a) the mapping \(f\) is \((S'_1;S'_2)\)-continuous at zero; b) the family of mappings \(x\to x+x_0\), \(x,x_0\in E_i\), \(i=1,2\), is \((S_i;S_i)\)-equicontinuous; c) the structure \(S_1\) is perfect.

Theorem 7. A linear \((S_1;S_2)\)-continuous mapping \(f\) from a semisyntopogenous vector space \([E_1;S_1]\) into a semisyntopogenous vector space \([E_2;S_2]\) is \(((S_1)_\pi;(S_2)_\pi)\)-continuous.

Theorem 8. Let \([E;S]\) be an \(n\)-dimensional semisyntopogenous vector space over a \(g\)-complete* syntopogenous field \([K;S']\). If every \(S\)-closed hypersubspace \(H\) of the space \([E;S]\) has a semi-

* That is, such that the space \([K;S'_\pi]\) is complete.

syntopogenous completion, and every one-dimensional subspace is isomorphic to the space \([K_s; S']\) (here \(K_s\) is the field \(K\), regarded as a vector space over itself), then the space \([E; S]\) is isomorphic to the space

\[ \left[\prod_{i=1}^{n} K_s^{\,i};\ \prod_{i=1}^{n} S^i\right], \quad \text{where } K_s^{\,i}=K_s,\ S^i=S',\ 1\le i\le n. \]

Theorem 9. Suppose that: a) \([X; S_2]\) is a barrelled \(\delta\)-space*; b) \([Y; S]\) is a locally convex \(R_3\)-regular syntopogenous space; c) the semisyntopogenous space \([L(X;Y); S_{\mathfrak{x}}' \mid L(X;Y)]\) is a semisyntopogenous vector space, where \(L(X;Y)\) is the set of linear \((S_2;S)\)-continuous mappings \(X\to Y\); moreover we assume that \(\mathfrak{x}=(X_\alpha)_{\alpha\in A}\) is the family of all possible finite parts of the set \(X\); d) the set \(H\subset L(X;Y)\) is bounded with respect to the structure \(S_{\mathfrak{x}}' \mid L(X;Y)\). Then the set \(H\) has the following property: if \(\{0\}<V\) for \(<\in S\), then \(\{0\}<_{1}^{-1}u(V)\) for some \(<_{1}\in S_2\) and for all mappings \(u\) from the set \(H\).

Example. On the set \(E\) of real numbers define the relation \(<_{n}\) as follows: \(A<_{n}B\) means that \(A\subset G\subset B\), where \(G\) is a set open in the classical topology of the number line, whose open bounded component intervals do not exceed \(n\) in length. It turns out that: 1) \(<_{n}\) is a semitopogenous order on the set \(E\); 2) \(S=\{<_{n};\, n=1,2,3,\ldots\}\) is a semisyntopogenous structure on \(E\); 3) \([E;S]\) is a semisyntopogenous, not syntopogenous (and, a fortiori, not topological!) group; 4) \([E;S]\) is a semisyntopogenous vector space distinct from every topological one.

Moscow State Pedagogical Institute
named after V. I. Lenin

Received
17 XI 1967

REFERENCES

1. Á. Császár, Grundlagen der allgemeinen Topologie, Budapest, 1963.

* By a \(\delta\)-space we mean a space in which the set of neighborhoods of each point forms a filter.