Mathematics

Z. Frolík

On an Internal Characterization of Topologically Complete Spaces in the Sense of E. Čech

(Presented by Academician P. S. Aleksandrov, 14 X 1960)

A space* \(P\) is called topologically complete in the sense of E. Čech (hereafter simply complete) if it is a \(G_\delta\)-set in its Čech compactification \(\beta(P)\). In the paper \((^2)\) the author succeeded, by means of so-called complete sequences of open covers, in giving an internal characterization of complete spaces. In the present note an internal characterization of complete spaces is given by means of a “diameter,” and also by means of a certain ordering of a system of open subsets.

Definition 1. By a diameter on a space \(P\) we shall mean any nonnegative function** \(d\), defined on the system \(\exp P\) of all subsets of the set \(P\) and satisfying the following conditions:

(d1) If \(M \subset N \subset P\), then \(d(M) \leq d(N)\).

(d2) For every \(M \subset P\), \(d(M)\) is the greatest lower bound of the set of all \(d(U)\), where \(U\) is an open set containing \(M\).

(d3) \(d((x))=0\) for every point \(x \in P\).

Example 1. If \(\varphi\) is a pseudometric on the space \(P\) and if \(d(M)\) denotes the diameter of the set \(M\) in the usual sense, i.e. \(d(M)=\sup \varphi(x,y)\), \(x \in M,\ y \in M\), then \(d\) is a diameter in the sense of our definition. We shall call it the diameter generated by the pseudometric \(\varphi\). If \(f\) is a continuous function on the space \(P\) and if \(d(M)\) denotes the oscillation of the function \(f\) on the set \(M\), then \(d\) is a diameter generated by the function \(f\).

Definition 2. Let \(d\) be a diameter on the space \(P\). A centered system \(\mathfrak A\) of subsets of the space \(P\) will be called \(d\)-shrinking (or a \(d\)-system) if \(\inf\{d(A);\ A \in \mathfrak A\}=0\). The diameter \(d\) will be called complete if the intersection of the closures of the sets of every \(d\)-system is nonempty.

The fundamental property of a complete diameter is its nonextendability. We shall call a space \(R\) an extension of the space \(P\) if \(R \supset P\) and \(\overline P=R\); if, in addition, \(R=P\), then \(R\) is called an identical extension of the space \(P\).

Definition 3. A diameter \(d\) on the space \(P\) will be called nonextendable if every extension \(R\) of the space \(P\) on which there exists a diameter \(D\) such that \(d(M)=D(M)\) for \(M \subset P\) is identical.

Theorem 1. In order that a diameter on a space be nonextendable, it is necessary and sufficient that it be complete.

Proof. Suppose that \(D\) is a diameter on a space \(R\), \(P \subset R\), \(\overline P=R\), \(P \ne R\), and \(d\) is the same function \(D\), considered on \(\exp P\). Let \(x \in R-P\). Then the system \(\mathfrak B\) of all \(A \cap P\), where

* “Space” throughout the paper means a completely regular topological space.

** By a “function” we shall always understand a real-valued function (which, generally speaking, also assumes the values \(-\infty\) and \(\infty\)).

\(A\)—a neighborhood of \(x\) in \(R\), is a \(d\)-system, and the intersection of the closures in \(P\) of the sets from \(\mathfrak{B}\) is empty. Thus, the diameter \(d\) is not complete.

Now let \(d\) be an incomplete diameter on the space \(P\). There exists a \(d\)-system \(\mathfrak{A}\) such that the intersection of the closures of the sets from \(\mathfrak{A}\) is empty. Take some point \(x\) lying in the intersection of the closures of these sets in the Čech extension \(\beta(P)\), and consider the space \(R=P\cap(x)\subset\beta(P)\). For every open \(U\subset R\), put \(D(U)=d(U\cap P)\), and for every \(M\subset R\) put

\[ D(M)=\inf\{D(U);\ U\supset M,\ U\ \text{open}\}. \]

It turns out that \(D\) is a diameter on the set \(R\). Clearly, \(d(M)=D(M)\) for \(M\subset P\).

Theorem 2. If there exists a complete diameter on the space \(P\), then \(P\) is a \(G_\delta\)-set in every extension of itself.

Proof. Let \(d\) be a complete diameter on the space \(P\), and let \(R\) be an extension of the space \(P\). For every natural \(n\), let \(U_n\) be the union of all open \(U\subset R\) satisfying the inequality \(d(P\cap U)<\frac1n\). Clearly, there exists an extension of the diameter \(d\) to \(\bigcap_{n=1}^{\infty}U_n\). Thus,

\[ \bigcap_{n=1}^{\infty}U_n=P. \]

Theorem 3. If there exists a complete diameter on the space \(P\), then there exists a complete diameter on every \(G_\delta\)-subset of the space \(P\).

Proof. Let \(d\) be a complete diameter on the space \(P\), and let

\[ R=\bigcap_{n=1}^{\infty}U_n, \]

where \(U_n\) is an open subset of the space \(P\). For every open \(U\subset R\), put

\[ d_1(U)=\inf\left\{\frac1n;\ \overline{U}\cap R\subset U_n\right\}, \]

\[ D(U)=\max[d(U),d_1(U)]. \]

For every \(M\subset R\), put

\[ D(M)=\inf\{D(U),\ U\ \text{open in }R,\ U\supset M\}. \]

It turns out that \(D\) is a complete diameter on the space \(R\).

Corollary. The space \(P\) is complete if and only if there exists a complete diameter on \(P\).

It is not difficult to prove the following two theorems.

Theorem 4. A diameter \(d\) on the space \(P\) is complete if and only if the following two conditions are satisfied:

(1) If \(d(M)=0\), then the closure of the set \(M\) is compact.

(2) The intersection of the closures of the sets of any countable \(d\)-system is nonempty.

Theorem 5. Let \(d\) be a complete diameter on the space \(P\). For every \(M\subset P\), let \(d_1(M)\) be the greatest lower bound of the set of all \(\varepsilon>0\) for which there exists a finite number of sets \(M_1,\ldots,M_k\) such that

\[ M\subset\bigcup_{i=1}^{k}M_i\quad\text{and}\quad d(M_i)<\varepsilon. \]

Then \(d_1\) is a complete diameter on the space \(P\).

A sequence \(\{\mathfrak{A}_n\}\) of open covers of the space \(P\) is called complete if, for every centered system of sets \(\mathfrak{A}\) such that \(\mathfrak{A}\cap\mathfrak{A}_n\ne\varnothing\) for all \(n=1,2,\ldots\), the intersection of the closures of the sets from \(\mathfrak{A}\) is nonempty.

Let \(\alpha=\{\mathfrak{A}_n\}\) be a sequence of open covers of the space \(P\) satisfying the following two conditions:

(p1) \(\mathfrak A_n \supset \mathfrak A_{n+1},\ n=1,2,\ldots\).

(p2) If \(A\) is open and \(A\subset B\in\mathfrak A_n\), then \(A\in\mathfrak A_n\).

For every open \(U\subset P\) put \(d(U)=1\), if \(U\notin\mathfrak A_1\). If, however, \(U\in\mathfrak A_1\), put
\[ d(U)=\inf\left\{\frac1n;\ U\in\mathfrak A_n\right\}. \]
Finally, for any \(M\subset P\) put
\[ d(M)=\inf\{d(U);\ U\supset M,\ U\text{ open}\}. \]
Obviously, \(d\) is a diameter on the space \(P\). It turns out that \(\alpha\) is a complete sequence if and only if \(d\) is a complete diameter.

Now let \(d\) be a diameter on the space \(P\). Let \(\mathfrak A_n\) be the system of all open sets \(U\) for which \(d(U)<\frac1n\). It turns out that the sequence \(\{\mathfrak A_n\}\) of open coverings is complete if and only if \(d\) is a complete diameter.

If a complete sequence of open coverings is given, it is not difficult to construct a complete sequence satisfying conditions (p1) and (p2). Thus, a complete sequence of open coverings exists if and only if a complete diameter exists.

In proofs it is convenient to use additive complete sequences satisfying conditions (p1) and (p2), since if \(\{\mathfrak A_n\}\) is a complete sequence and if \(\mathfrak B_n\) consists of the unions of all finite subsystems of the covering \(\mathfrak A_n\), then \(\{\mathfrak B_n\}\) is also a complete sequence.

Let \(d\) be a complete diameter on the space \(P\). Define an ordering of open sets \(>\) so that \(A>B\) if and only if \(A\supset B\) and
\[ 2d(B)\leqslant \min[d(A),1]. \]
It turns out that the following four conditions are fulfilled:

(u1) If \(A>B\), then \(A\supset B\).

(u2) If \(C\) and \(D\) are open, \(C\supset A\), \(D\subset B\), and \(A>B\), then \(C>D\).

(u3) If \(A\) is an open set, then the system of all \(B<A\) is a basis for the open subsets of the set \(A\).

(u4) If \(\mathfrak A\) is a centered system of subsets of the space \(P\) and if for every natural number \(n\) there exist \(A_1,\ldots,A_{n+1}\) such that \(A_i>A_{i+1}\), \(i=1,\ldots,n\), and \(A_{n+1}\) contains some \(A\in\mathfrak A\), then the intersection of the closures of the sets from \(\mathfrak A\) is nonempty.

Now let \(>\) be some ordering of open sets satisfying conditions (u1)—(u4). For every \(M\subset P\) let \(d(M)\) be the lower bound of all \(\frac1n\) for which there exist \(A_1,\ldots,A_{n+1}\) such that \(A_{n+1}\supset M\) and \(A_i>A_{i+1}\), \(i=1,\ldots,n\). If no such \(n\) exists, put \(d(M)=1\). It turns out that \(d\) is a complete diameter on the space \(P\).

Thus the following is true:

Theorem 6. The following properties of the space \(P\) are equivalent:

(1) \(P\) is a complete space.

(2) \(P\) is a \(G_\delta\)-set in some complete space.

(3) \(P\) is a \(G_\delta\)-set in every one of its extensions.

(4) There exists a complete diameter on the space \(P\).

(5)\(^*\) There exists a complete sequence of open coverings of the space \(P\).

\(^*\) The equivalence of completeness to property (5) was proved in the work of A. Arhangel’skii \((^3)\), submitted for publication on 18 V 1960. P. S. Aleksandrov.

(6) There exists an ordering of the open sets of the space satisfying conditions (u1)—(u4).

Theorem 7. The following properties of a space are equivalent:

(1) There exists a perfect* mapping of the space \(P\) onto some complete metric space.

(2) There exists on \(P\) a complete diameter generated by a pseudometric.

(3) \(P\) is a complete space and every additive open covering of the space \(P\) is normal.**

Charles University
Prague, Czechoslovakia

Received
14 X 1960

References

\({}^{1}\) E. Čech, Ann. of Math., 38, 823 (1937).
\({}^{2}\) Z. Frolík, Czechoslovak Math. J., 85, 359 (1960).
\({}^{3}\) A. Arhangel’skii, Vestn. Mosk. Univ., Ser. Math. and Mech., No. 1 (1961).
\({}^{4}\) Z. Frolík, Czechoslovak Math. J., 84, 63 (1959).

* A continuous and closed mapping is called perfect if the full preimages of all points are bicompact.

** A covering \(\mathfrak A\) is called normal if there exists a sequence \(\{\mathfrak A_n\}\) of open coverings such that \(\mathfrak A_1\) is inscribed in \(\mathfrak A\) and \(\mathfrak A_{n+1}\) is star-inscribed in \(\mathfrak A_n\), \(n = 1, 2, \ldots\).