I. M. Dektyarev

On (\varphi)-Complete Spaces Metrized over a Semifield

(Presented by Academician P. S. Aleksandrov on 17 VII 1963)

In note ((^3)), for a space metrized over a semifield, the notion of (\varphi)-completeness was introduced and it was proved that every (\varphi)-complete space is a space of the second category in itself. A slight modification of the proof makes it possible, from the same assumptions, to obtain a stronger conclusion. The present note is devoted to the exposition of this question.

Let us first recall the notion of a metric space over a semifield. Let (E) be some semifield and (K_E) the set of all its positive elements. A set (X) is called a metric space over the semifield (E) if a mapping (called a metric) is given:

[
\rho: X \times X \to K_E,
]

satisfying the following conditions:

1. (\rho(x,y)=0) if and only if (x=y).
2. (\rho(x,y)=\rho(y,x)).
3. (\rho(x,y)+\rho(y,z)\ge \rho(x,z)).

If (U) is an arbitrary neighborhood of zero in the metrizing semifield (E) and (x\in X), then by (\Omega(x,U)) we shall denote the set of all those elements (y\in X) for which (\rho(x,y)\in U). The collection of all sets of the form (\Omega(x,U)) may be taken as a base of neighborhoods in (X) (see ((^1)), p. II, 1). The topology obtained in this way is called the natural topology of the metric space (X). A topological space (X) is called metrized over a semifield if a metric is introduced in it and the natural topology of the metric space (X) coincides with the original topology of the space (X).

Let us recall the basic definitions of ((^3)). Let (\Delta) be some infinite set; (\Omega) the set of all its finite subsets, partially ordered by inclusion; (\varphi) a mapping of the set (\Omega\setminus\Delta) into (\Omega), satisfying the following conditions: 1) (\varphi(\xi)<\xi) for all (\xi\in\Omega\setminus\Delta); 2) whatever (\xi_1,\xi_2\in\Omega) may be, there is an element (\xi_3\in\Omega) and a natural number (k) such that the relations (\xi_3>\xi_2) and (\varphi^k(\xi_3)=\xi_1) hold. The proof of existence and the construction of such a mapping (\varphi) are given in ((^2)).

Definition 1. We shall say that (\xi_1) is a (\varphi)-larger element than (\xi_2), and denote this by

[
\xi_1 \overset{\varphi}{>} \xi_2,
]

if there exists a natural number (k) such that (\varphi^k(\xi_1)=\xi_2).

Definition 2. A sequence of type (\Omega) of points of a topological space (X), metrized over (R_\Delta), will be called (\varphi)-fundamental if for every neighborhood of zero (U) in (R_\Delta) there exists (\xi_0\in\Omega) such that from

[
\xi_1>\xi_0,\qquad \xi_2 \overset{\varphi}{>} \xi_1
]

it follows that (\rho(x_{\xi_1},x_{\xi_2})\in U).

Definition 3. A point (a\in X) will be called a (\varphi)-limit for a sequence of type (\Omega) if for every neighborhood of zero (U) in (R_\Delta) and every (\xi_0\in\Omega) there exists an element

[
\xi \overset{\varphi}{>} \xi_0,
]

such that (\rho(x_\xi,a)\in U).

Definition 4. We shall call the space (X) (\varphi)-complete if, for every (\varphi)-fundamental sequence ({x_\xi}), there exists at least one (\varphi)-limit point.

Theorem 1. Let the space (X) be metrized over (R_\Delta), and suppose that for some mapping (\varphi) satisfying the conditions 1) and 2) mentioned above, the space (X) is (\varphi)-complete. Then the space (X) cannot be represented as the sum of (m) nowhere dense subsets, where (m) is the cardinality of the set (\Delta) (or, what is the same thing, the intersection of open everywhere dense subsets of the space (X) is nonempty).

Proof. Let (\xi={q_1,\ldots,q_n}) be an arbitrary element of the set (\Omega). By (U_\xi) we shall denote the neighborhood of zero in (R_\Delta) consisting of functions which take, at (q_i\in\xi), values whose moduli are less than (2^{-|\xi|}), where (|\xi|) is the number of elements in (\xi). Let ({G_q}) be an arbitrary family of open everywhere dense subsets in (X) (the index ranges over the set (\Delta)).

We construct, by induction on (\xi), such a (\varphi)-fundamental sequence ({x_\xi}) that

[
x_\xi \in \bigcap_{q\in\xi} G_q.
]

Simultaneously by induction we construct a collection of such neighborhoods (V_\xi) that (x_\xi\in V_\xi), (x_{\varphi(\xi)}\in V(\xi)), and

[
\overline V(\xi)\subset V_{\varphi(\xi)}\cap
\left(\bigcap_{q\in\varphi(\xi)} G_q\right).
]

If (\xi=q), then as (x_\xi) we take any point of the set (G_q). For (V_\xi) for these (\xi) we take (G_q).

Suppose now that all (x_\xi) and (V_\xi) for which (|\xi|1), have been constructed, and let (|\xi_0|=k). Since (\varphi(\xi_0)<\xi_0), we have (|\varphi(\xi_0)|<k), and hence, by the induction hypothesis, the point (x_{\varphi(\xi_0)}) has already been constructed.

Take such a neighborhood (V_{\xi_0}) of the point (x_{\varphi(\xi_0)}) that

[
\overline V_{\xi_0}\subset
V_{\varphi(\xi_0)}\cap
\left(\bigcap_{q\in\varphi(\xi_0)}G_q\right)\cap
\Omega(x_{\varphi(\xi_0)},U_{\xi_0}).
]

The intersection of the constructed neighborhood (V_{\xi_0}) with (\bigcap_{q\in\xi_0}G_q) is nonempty, since the sets (G_q) are everywhere dense in (X). Take an arbitrary point in the intersection

[
V_{\xi_0}\cap\left(\bigcap_{q\in\xi_0}G_q\right)
]

and denote it by (x_{\xi_0}). The induction carried out makes it possible to construct (x_\xi) and (V_\xi) for all (\xi\in\Omega).

We prove that the sequence obtained is (\varphi)-fundamental. Let (U) be some neighborhood of zero in (R_\Delta). There exists an element (\xi_0\in\Omega) such that (U_{\xi_0}\subset U) (see (1)). Let now (\xi_1>\xi_0), (\xi_2\overset{\varphi}{>}\xi_1), and let (k) be such a natural number that (\varphi^k(\xi_2)=\xi_1). Then

[
\rho(x_{\xi_2},x_{\xi_1})\le
\rho(x_{\xi_2},x_{\varphi(\xi_2)})+
\rho(x_{\varphi(\xi_2)},x_{\varphi^2(\xi_2)})+\cdots+
\rho(x_{\varphi^{k-1}(\xi_2)},x_{\xi_1}).
]

But

[
\rho(x_{\varphi^i(\xi_2)},x_{\varphi^{i+1}(\xi_2)})\in
U_{\varphi^i(\xi_2)}\subset
\frac12 U_{\varphi^{i+1}(\xi_1)}
\subset\cdots\subset
\frac{1}{2^{k-i}}U_{\varphi^k(\xi_2)}
=
\frac{1}{2^{k-i}}U_{\xi_1}
]

(recall that (|\varphi(\xi)|\le|\xi|-1)). Thus, (\rho(x_{\varphi^i(\xi_2)},x_{\varphi^{i+1}(\xi_2)})) at (q_l\in\xi_1) assumes values less than

[
\frac{1}{2^{k-i}}\frac{1}{2^{|\xi_1|}},
]

and consequently (\rho(x_{\xi_2},x_{\xi_1})) assumes at these (q_l) values less than

[
\frac{1}{2^{|\xi_1|}}\sum_{i=0}^{k-1}\frac{1}{2^{k-1}}<
\frac{1}{2^{|\xi_1|}}.
]

Hence,

[
\rho(x_{\xi_2},x_{\xi_1})\in U_{\xi_1}\subset U_{\xi_0}.
]

Thus, the sequence ({x_\xi}) is (\varphi)-fundamental. By virtue of the (\varphi)-completeness of the space (X), it has a (\varphi)-limit point (a). Let (U) be an arbitrary neighborhood of zero in (R_\Delta). For any (\xi_0 \in \Omega) there is an element (\xi) such that (\xi \overset{\varphi}{>} \xi_0) and (\rho(x_\xi,a)\in U). But by the construction of the sequence ({x_\xi}) and the fact that (\xi \overset{\varphi}{<} \xi_0), we have

[
x_\xi \in V_\xi \subset V_{\xi_0},
]

and therefore

[
a \in V_{\xi_0} \subset \bigcap_{q\in\varphi(\xi_0)} G_q.
]

Since this is true for any element (\xi_0\in\Omega), and since for any element (q\in\Delta) there exists such a (\xi_0) that (q\in\varphi(\xi_0)) (see property 2 of the mapping (\varphi)), it follows that (a\in \bigcap_{q\in\Delta} G_q), as was required to prove.

Tashkent State University
named after V. I. Lenin

Received
5 VII 1963

REFERENCES

¹ M. Ya. Antonovskii, V. G. Boltyanskii, T. A. Sarymsakov, Topological semifields, Tashkent, 1960. ² M. Ya. Antonovskii, V. G. Boltyanskii, T. A. Sarymsakov, Proceedings of Tashkent State University, 1961, p. 191. ³ I. M. Dektyarev, Reports of the Academy of Sciences of the Uzbek SSR, No. 6 (1962).