UDC 519.50+519.54

MATHEMATICS

Kh. N. Inasaridze

ON ONE GENERALIZATION OF PERFECT MAPPINGS

(Presented by Academician P. S. Aleksandrov, September 3, 1965)

We shall say that a space \(X\) (Hausdorff and completely regular) or a mapping \(f\) (continuous) has property \(P\) at the \(n\)-th infinity (or simply at infinity, when \(n=1\)) if the Čech growth of order \(n\) of the space \(X\) or of the mapping \(f\) has property \(P\) (the definition of the growth of order \(n\) of a space or mapping is given in \((^3)\)). We shall denote the growth of order \(n\) of a mapping \(f\) by \(c^n(f)\), where \(c^0(f)=f\), and the Čech growth of order \(n\) by \(\gamma^n(f)\)*.

A mapping \(f\) of a space \(X\) into a space \(Y\) will be called a \(K_n\)-mapping if the inverse image \(f^{-1}(y)\) of each point \(y\in Y\) is a space of the class \(K_n\) (the definition of the class \(K_n\) is given in \((^3)\)). A compact mapping is a \(K_0\)-mapping, and a \(K_1\)-mapping may naturally be called a locally compact mapping.

A mapping \(f\) of a space \(X\) onto a space \(Y\) will be called \(n\)-perfect if the Čech growths \(\gamma^0(f), \gamma^1(f), \ldots, \gamma^n(f)\) of the mapping \(f\) are closed mappings and \(f\) is a \(K_n\)-mapping. It is clear that a 0-perfect mapping is a perfect mapping (see \((^1),(^2)\)), while a closed mapping perfect at infinity (see \((^3)\)) is a 1-perfect mapping.

Using the results of \((^3)\), one can prove the following theorem.

Theorem 1. A mapping of a space from \(K_n\) onto an arbitrary space is a \(K_n\)-mapping and is perfect at the \(n\)-th infinity.

Because of lack of space, the proofs of this and of the remaining theorems are not given.

Theorem 2. A 1-perfect mapping \(X\) onto \(Y\) is a closed mapping, perfect at infinity.

Theorem 2 is proved with the aid of the following lemma.

Lemma 1. If \(f\) is a closed mapping of \(X\) into \(Y\), and \(\bar f:bX\to bY\) is an extension of \(f\) to compact extensions \(bX\) and \(bY\), then the equality

\[ \bar f^{-1}(y)\cap(bX\setminus X)=\bar f^{-1}(y)\setminus f^{-1}(y) \]

holds for every \(y\in Y\).

From this lemma, in particular, Theorem 4 of \((^3)\) also follows.

With the aid of Lemma 1 one can prove the following theorem.

Theorem 3. A closed mapping \(f\) of a space \(X\) onto a space \(Y\) is a \(K_n\)-mapping if and only if \(f\) is a \(K_{n-1}\)-mapping at infinity.

Using Theorems 2 and 3, we obtain the following theorem.

Theorem 4. Let \(f\) be a mapping of \(X\) onto \(Y\) such that the mappings \(\gamma^0(f), \gamma^1(f), \ldots, \gamma^n(f)\) are closed. Then \(f\) is an \(n\)-perfect mapping if and only if \(f\) is perfect at the \(n\)-th infinity.

In particular, for \(n=1\) we obtain Theorem 7 of \((^3)\).

* “A space or mapping has property \(P\) at the zeroth infinity” means that the space or mapping has property \(P\).

With the aid of Lemma 1 one can prove the following theorem.

Theorem 5. Let \(f\) be a mapping of \(X\) onto \(Y\) such that the mappings
\(\gamma^0(f), \ldots, \gamma^n(f)\) are closed. Then the mappings
\(c^0(f), \ldots, c^n(f)\) are closed, and \(R[\gamma^i(f)]\) is a closed subset of
\(R[\gamma^{i-1}(f)]\), where \(1 \le i \le n\)*.

The following theorems follow from Theorem 5.

Theorem 6. Let \(f\) be a mapping of \(X\) onto \(Y\) such that the mappings
\(\gamma^0(f), \ldots, \gamma^{n-1}(f)\) are closed. Then, if some extension of order \(n\) of the mapping \(f\) is perfect, all extensions of order \(n\) are perfect.

Theorem 7. Let \(f\) be a mapping of \(X\) onto \(Y\) such that the mappings
\(\gamma^0(f), \ldots, \gamma^{n-1}(f)\) are closed. Then \(f\) is an \(n\)-perfect mapping if and only if, for every point
\(x \in R^{(n+1)/2}(X)\), the set
\(f^{-1}[f(x)] \cap R^{(n-1)/2}(X)\) is compact, if \(n\) is odd, and the mapping induced by the mapping \(f\) on \(R^{n/2}(X)\) is perfect, if \(n\) is even.

In particular, for \(n=1\) we obtain Theorem 6 from \((^3)\).

Let \(f\) be a mapping of \(X\) onto \(Y\) and \(g\) a mapping of \(Y\) onto \(Z\). It is easy to see that, if \(f\) is a perfect mapping, then the mapping
\(fg: X \to Z\) is a \(K_n\)-mapping if and only if \(g\) is a \(K_n\)-mapping.

Using Theorems 5 and 8 from \((^3)\), one can prove the following theorems.

Theorem 8. Let \(f\) be a closed mapping such that \(R(f)\) is a space in \(K_n\). Then, if \(fg\) is a \(K_1\)-mapping, \(g\) is a \(K_{n+2}\)-mapping; and if \(fg\) is a \(K_2\)-mapping, then for even \(n\), \(g\) is a \(K_{n+2}\)-mapping, while for odd \(n\), \(g\) is a \(K_{n+3}\)-mapping.

Theorem 9. Let \(f\) be a closed, closed at infinity, locally compact mapping, and let \(R(f)\) be compact. Then: 1) if \(g\) is a \(K_n\)-mapping, then for even \(n\), \(fg\) is a \(K_{n+1}\)-mapping, while for odd \(n\), \(fg\) is a \(K_n\)-mapping; 2) if \(fg\) is a \(K_n\)-mapping, then for even \(n\), \(g\) is a \(K_n\)-mapping, while for odd \(n\), \(g\) is a \(K_{n+1}\)-mapping.

With the aid of Theorems 4 and 7 we obtain the following theorem.

Theorem 10. Let \(f\) be an \(n\)-perfect mapping of \(X\) onto \(Y\), where \(n\) is even. Then, if \(Y\) is paracompact, finally compact, or a space in \(K_m\), then \(X\) is, in the \(n\)-th infinity, respectively paracompact, finally compact, or a space in \(K_m\).

By virtue of one result from \((^3)\), it follows that if some extension of order \(n\) of a space has the perfect property, then all extensions of order \(n\) of this space have the same property. Using this fact and Theorem 4 from \((^3)\), one can prove the following theorem.

Theorem 11. If \(f\) is a closed mapping of a locally compact space \(X\) onto a space \(Y\), and \(R(f)\) is compact, paracompact, or finally compact in the \(n\)-th infinity, then \(Y\) is, in the \((n+2)\)-th infinity, respectively compact, paracompact, or finally compact.

With the aid of Lemma 17 of E. G. Sklyarenko from \((^4)\) (see also Theorem 3.6 from \((^5)\)) we obtain the following theorem.

Theorem 12. The space \(X\) is finally compact in the \(n\)-th infinity if and only if \(R^{n/2}(X)\) is finally compact, if \(n\) is even, and \(R^{(n-1)/2}(X)\) is a space of type \(\mathfrak{S}\), if \(n\) is odd**.

* If \(f\) is a mapping of \(X\) into \(Y\), then \(R(f)\) is the set of all points \(y \in Y\) such that \(f^{-1}(y)\) is noncompact (see \((^3)\)).

** The definition of a space of type \(\mathfrak{S}\) is given in \((^4)\).

We note that a finally compact or paracompact space is, in any even infinity, finally compact or paracompact, respectively.

Theorem 13. Let \(f\) be a closed mapping of \(X\) onto \(Y\), and suppose that each point of \(R(f)\) has a compact neighborhood. Then, if \(X\) is locally compact or is a space of type \(\mathfrak{S}\), then \(Y\) is locally compact or a space of type \(\mathfrak{S}\), respectively.

Theorem 14. Let \(f\) be a closed mapping of \(X\) onto \(Y\), perfect at infinity, and suppose that \(R(f)\) is finally compact. Then, if \(Y\) is a space of type \(\mathfrak{S}\), then \(X\) is a space of type \(\mathfrak{S}\).

Theorem 15. Let \(f\) be a closed mapping of \(X\) onto \(Y\), closed at infinity, and suppose that each point of \(R(f)\) has a compact neighborhood. Then, if \(X\) is paracompact at infinity, then \(Y\) is also paracompact at infinity.

We give one more theorem, generalizing Theorem 1 of E. G. Sklyarenko from \((^{6})\) on the extension of perfect mappings to compact extensions.

Theorem 16. Let \(bX\) and \(bY\) be compact extensions of the spaces \(X\) and \(Y\), with \(bX\) perfect and \(bY\) having a punctiform growth. Every closed mapping \(f\) of the space \(X\) into \(Y\) such that \(R(f)\) is compact,

\[ \dim R(f)=0 \]

and each point of \(R(f)\) has a compact neighborhood in \(\overline{f(X)}\), extends to a mapping \(\bar f : bX \to bY\).

This theorem is proved with the aid of Lemma 1.

Tbilisi Mathematical Institute
named after A. M. Razmadze
Academy of Sciences of the Georgian SSR

Received
20 VIII 1965

CITED LITERATURE

\(^{1}\) P. S. Aleksandrov, UMN, 15, issue 2, 25 (1960).
\(^{2}\) V. I. Ponomarev, Mat. sborn., 51, No. 4, 515 (1960).
\(^{3}\) Kh. N. Inasaridze, DAN, 166, No. 5 (1966).
\(^{4}\) E. G. Sklyarenko, Izv. AN SSSR, ser. matem., 26, 427 (1962).
\(^{5}\) Melvin Henriksen, J. R. Isbell, Duke Math. J., 25, No. 1, 83 (1958).
\(^{6}\) E. G. Sklyarenko, DAN, 146, No. 5, 1031 (1962).