MATHEMATICS

V. KLYUSHIN

ON PERFECT MAPPINGS OF PARACOMPACT SPACES

(Presented by Academician P. S. Aleksandrov on 4 VI 1964)

In the present paper we investigate properties of paracompact spaces admitting perfect mappings* onto metric spaces, and consider the question of approximating spaces by inverse spectra of metric spaces with perfect projections. The results obtained here adjoin the results of V. Ponomarev \((^{1,7})\), B. Pasynkov \((^{2})\), and A. Arkhangel’skii \((^{8})\).

B. Pasynkov, generalizing the well-known theorems of S. Mardešić \((^{3})\), proved in \((^{2})\) a theorem on the factorization of continuous mappings into metric spaces. For perfect mappings the following assertion holds.

Theorem 1. If for a paracompact space \(X\) there exists a perfect mapping \(f: X \to R\) onto some metric space \(R\), then for every cover \(\omega\) of the space \(X\) there exist a metric space \(S\) and perfect mappings \(g: X \to S\) and \(h: S \to R\) such that \(g\) is an \(\omega\)-mapping* and \(f = hg\). If, moreover, \(\dim X = n\), then for every cover \(\omega\) of the space \(X\) there exist a metric space \(\widetilde S\), whose dimension \(\dim \widetilde S = n\), and perfect mappings \(\widetilde g: X \to \widetilde S\) and \(\widetilde h: \widetilde S \to R\), subject to the same conditions: \(\widetilde g\) is an \(\omega\)-mapping and \(f = \widetilde h \widetilde g\).

For the proof we shall need the following

Lemma. Let a perfect mapping \(f: X \to R\) of a completely regular space \(X\) onto a completely regular space \(R\) be given, a mapping \(g: X \to S\) onto a completely regular space \(S\), and a mapping \(h: S \to R\). Then, if \(f = hg\), the mappings \(g\) and \(h\) are also perfect.

Proof of the lemma. It is easy to see that the mapping \(h\) is perfect under our assumptions. The mapping \(g\) is bicompact, since the elements of the decomposition \(\{g^{-1}z\}_{z \in S}\) are closed subsets of the elements of the decomposition \(\{f^{-1}y\}_{y \in R}\). To prove the lemma it remains only to prove the closedness of the mapping \(g\). Consider the mappings
\[ \overline f: \beta X \to \beta R,\quad \overline g: \beta X \to \beta S \quad \text{and} \quad \overline h: \beta S \to \beta R, \]
which are extensions of the mappings \(f\), \(g\), and \(h\) to the maximal bicompact extensions \(\beta X\) and \(\beta S\), so that \(\overline f(\beta X \setminus X)=\beta R \setminus R\), \(\overline h(\beta S \setminus S)=\beta R \setminus R\)**** and \(\overline f=\overline h\,\overline g\). To prove the closedness of the mapping \(g\), it is enough to show that \(\overline g(\beta X \setminus X)=\beta S \setminus S\). Suppose that this last equality does not hold. Then, since \(\overline h(\beta S \setminus S)=\beta R \setminus R\) and \(\overline f=\overline h\,\overline g\), the equality
\[ \overline f(\beta X \setminus X)=\beta R \setminus R \]
would not hold. The lemma is proved.

Proof of Theorem 1. Take in the space \(R\) a sequence \(\{\alpha_i\}\) of covers by balls of radius \(1/2^i\), \(i=1, 2, \ldots\). The sequence of covers \(\{\omega_i\}\) of the space \(X\), where \(\omega_i=f^{-1}\alpha_i\),

* A continuous mapping \(f: X \to R\) is called perfect if it is closed and bicompact in the sense that the inverse images \(f^{-1}y\) of all points \(y \in R\) are bicompact.

** Covers here are everywhere assumed to be open.

*** A continuous mapping \(f: X \to S\) of a space \(X\) into some space \(S\) is called an \(\omega\)-mapping for a given cover \(\omega\) of the space \(X\) if for every point \(y \in S\) there exists a neighborhood \(Oy\) whose inverse image \(f^{-1}Oy\) is contained in at least one element of the cover \(\omega\).

**** These relations are fulfilled automatically, since \(f\) and \(h\) are perfect.

\(i=1,2,\ldots\), is a normal sequence.* Introduce the following notation. For any two covers \(\alpha\) and \(\beta\), denote by \(\alpha \wedge \beta\) the cover consisting of all sets that are intersections of some element of the cover \(\alpha\) with some element of the cover \(\beta\).

We shall now construct by induction a normal sequence of covers \(\{\gamma_i\}\), subject to the following conditions: 1) the cover \(\gamma_1\) is inscribed in the cover \(\omega\), and 2) the cover \(\gamma_i\) is inscribed in the cover \(\omega_i\) for each \(i=1,2,\ldots\). We do this as follows. For \(i=1\), take the cover \(\omega \wedge \omega_1\) as the cover \(\gamma_1\). Suppose that for all \(i=1,2,\ldots,k-1\) covers \(\gamma_i\) have been constructed satisfying our conditions, and as the cover \(\gamma_k\) take any cover star-inscribed in \(\gamma_{k-1}\wedge \omega_k\). Such a cover always exists, since \(X\) is paracompact.

The normal sequence \(\{\gamma_i\}\) induces some metric space \(S\) and a continuous mapping \(g:X\to S\) (see (1)). The sequence \(\{\gamma_i\}\) is inscribed in the sequence \(\{\omega_i\}\), and the elements of the partition \(\{g^{-1}z\}_{z\in S}\) are subsets of the elements of the partition \(\{f^{-1}y\}_{y\in R}\).

Define the perfect mapping \(h:S\to R\) as follows:
\[ h=fg^{-1}. \]
According to the lemma, the mapping \(g:X\to R\) is perfect.

Let now \(\dim X=n\). Into the normal sequence \(\{\gamma_i\}\), as shown by B. A. Pasynkov, one can inscribe a normal sequence of covers \(\{\mu_i\}\) subject to the following conditions:

\(1^\circ\). Each element of the cover \(\mu_{i+1}\), \(i=1,2,\ldots\), intersects only a finite number of elements of the cover \(\mu_i\).

\(2^\circ\). The multiplicity of any cover \(\mu_i\) does not exceed \(n+1\).

The space \(\widetilde S\) induced by the sequence of covers \(\{\mu_i\}\) has dimension \(\dim \widetilde S=n\) (see (6)). The mapping \(\widetilde g\), induced by this sequence, will, as is easy to verify, be perfect. The mapping \(\widetilde h=\widetilde f\widetilde g^{-1}\) is also perfect. The theorem is proved.

Theorem 2. A space \(X\) is a Čech-complete paracompactum if and only if for every cover \(\omega\) of the space \(X\) there exists a perfect \(\omega\)-mapping \(f:X\to S\) onto some complete metric space \(S\).

Proof.

\(1^\circ\). Necessity. Z. Frolík proved (5) that for a Čech-complete paracompactum there exists a perfect mapping onto a complete metric space. Taking an arbitrary cover \(\omega\) of the space \(X\) and applying Theorem 1, we obtain a metric space \(S\) and a perfect \(\omega\)-mapping \(f:X\to S\) onto the space \(S\). The completeness of the space \(S\) follows from a result proved by V. Ponomarev (4), asserting the preservation of completeness in both directions under perfect mappings.

\(2^\circ\). Sufficiency follows directly from the fact that if for every cover \(\omega\) of the space \(X\) there exists an \(\omega\)-mapping onto some metric space \(S\), then \(X\) is a paracompactum, and from the aforementioned result of V. Ponomarev. The theorem is proved.

Theorem 3. If a paracompactum \(X\) admits a perfect mapping onto some metric space \(R\), then \(X\) is the limit of some spectrum,
\[ X=\lim_{\leftarrow}\{X_\alpha,\mathfrak D_\alpha^\beta\}, \]
whose elements \(X_\alpha\) are metric spaces, and all projections \(\mathfrak D_\alpha^\beta\) are perfect mappings.

* A sequence of covers \(\{\omega_i\}\), \(i=1,2,\ldots\), is called normal if for each \(i\) the cover \(\omega_{i+1}\) is star-inscribed in the cover \(\omega_i\), i.e., for each point \(x\in X\) the union of the elements of the cover \(\omega_{i+1}\) containing this point is contained in some element of the cover \(\omega_i\).

Proof. Let \(\{\omega_\alpha\}_{\alpha\in\mathfrak A}\) be the set of all covers of the space \(X\). To each cover \(\omega_\alpha\) we assign a certain metric space \(X_\alpha\) and a mapping \(\widetilde{\omega}_\alpha:X\to X_\alpha\) which is a perfect \(\omega\)-mapping. Denote by \(D\) the diagonal of the product \(X\times X\). For each pair \(\alpha_1,\alpha_2\) take the space
\[ \overline{\omega}_{(\alpha_1\alpha_2)}D=\{\widetilde{\omega}_{\alpha_1}(x),\widetilde{\omega}_{\alpha_2}(x)\}=X_{(\alpha_1\alpha_2)}\subset X_{\alpha_1}\times X_{\alpha_2}. \]
Denote by \(h\) the natural mapping of the space \(X\) onto the space \(D\), which assigns to each point \(x\in X\) the point \((x,x)\in D\). The mapping
\(\widetilde{\omega}_{(\alpha_1\alpha_2)}=\overline{\omega}_{(\alpha_1\alpha_2)}h\), according to the lemma, is a perfect mapping of the space \(X\) onto the space \(X_{(\alpha_1\alpha_2)}\). The natural projections \(\widetilde{\omega}_{\alpha_1}^{(\alpha_1\alpha_2)}\) and \(\widetilde{\omega}_{\alpha_2}^{(\alpha_1\alpha_2)}\) of the space \(X_{(\alpha_1\alpha_2)}\) onto the factor spaces \(X_{\alpha_1}\) and \(X_{\alpha_2}\) are also perfect mappings; \(\widetilde{\omega}_{\alpha_1}=\widetilde{\omega}_{\alpha_1}^{(\alpha_1\alpha_2)}\widetilde{\omega}_{(\alpha_1\alpha_2)}\) and \(\widetilde{\omega}_{\alpha_2}=\widetilde{\omega}_{\alpha_2}^{(\alpha_1\alpha_2)}\widetilde{\omega}_{(\alpha_1\alpha_2)}\).

Now denote by \(D^{(2)}\) the diagonal of the product \(D\times D\). For each pair \((\alpha_1\alpha_2)\), \((\alpha_3\alpha_4)\) put
\[ \overline{\omega}_{(\alpha_1\alpha_2\alpha_3\alpha_4)}D^{(2)} =\{\widetilde{\omega}_{(\alpha_1\alpha_2)}(x),\widetilde{\omega}_{(\alpha_3\alpha_4)}(x)\} =X_{(\alpha_1\alpha_2\alpha_3\alpha_4)}\subset X_{(\alpha_1\alpha_2)}\times X_{(\alpha_3\alpha_4)} \]
and repeat the arguments given above.

Continuing this process, we obtain a spectrum
\(S=\{X_\alpha,\widetilde{\omega}_\alpha^\beta\}\), whose elements \(X_\alpha\) are metric spaces, and all projections \(\widetilde{\omega}_\alpha^\beta\) are perfect mappings (here the indices \(\alpha\) denote sets \((\alpha_1\alpha_2\ldots\alpha_{2^n})\)).

The mappings \(\widetilde{\omega}_\alpha:X\to X_\alpha\), defined for all indices \(\alpha=(\alpha_1\ldots\alpha_{2^n})\) and satisfying the condition \(\widetilde{\omega}_\alpha=\widetilde{\omega}_\alpha^\beta\widetilde{\omega}_\beta\) for \(\beta>\alpha\), determine a mapping \(\widetilde{\omega}\) of the space \(X\) into the space
\[ \widetilde X=\lim_{\leftarrow}\{X_\alpha,\widetilde{\omega}_\alpha^\beta\} \]
by assigning to each point \(x\in X\) the point \(\widetilde{\omega}(x)=\{x_\alpha\}\), where \(x_\alpha=\widetilde{\omega}_\alpha(x)\). We shall show that the mapping \(\widetilde{\omega}:X\to\widetilde X\) is a homeomorphism. Let \(x'\) and \(x''\) be two distinct points of the space \(X\). For the cover \(\omega=\{X\setminus x',\,X\setminus x''\}\) there exists \(\alpha=\alpha(x',x'')\) for which \(\widetilde{\omega}_\alpha:X\to X_\alpha\) is an \(\omega\)-mapping. It is clear that \(\widetilde{\omega}_\alpha(x')\ne\widetilde{\omega}_\alpha(x'')\). Thus the mapping \(\widetilde{\omega}:X\to\widetilde X\) is one-to-one. We now show that for every open set \(O\subset X\) the set \(\widetilde{\omega}(O)\) is open in \(\widetilde X\). Take an arbitrary point \(x\in O\). For the cover \(\omega=\{O,\,X\setminus x\}\) there exists an \(\alpha\) such that the mapping \(\widetilde{\omega}_\alpha:X\to X_\alpha\) will be an \(\omega\)-mapping. Hence there exists a neighborhood \(O_{\alpha y}\) of the point \(y=\widetilde{\omega}_\alpha(x)\) such that \(\widetilde{\omega}_\alpha^{-1}(O_{\alpha y})\subseteq O\). Running through all points of the set \(O\), we obtain that the set
\[ \widetilde{\omega}(O)=\widetilde{\omega}\Bigl(\bigcup_\alpha \widetilde{\omega}_\alpha^{-1}(O_{\alpha y})\Bigr) =\bigcup_\alpha \widetilde{\omega}_\alpha^{-1}(O_{\alpha y}), \]
where by \(\widetilde{\omega}_\alpha\) are denoted the projections of the space \(\widetilde X\) onto \(X_\alpha\). The mapping \(\widetilde{\omega}\) is open. Moreover, as is clear from the construction, the mapping \(\widetilde{\omega}\) is a mapping onto the whole space \(X\). The theorem is proved.

Since both paracompactness and completeness in the sense of Čech are preserved in both directions under perfect mappings, Theorem 3 immediately implies

Theorem 4. A space \(X\) is paracompact and complete in the sense of Čech if and only if it is the limit of some spectrum of complete metric spaces with perfect projections.

We now consider the question of approximating \(n\)-dimensional, in the sense of \(\dim\), paracompacts by spectra of \(n\)-dimensional, in the sense of \(\dim\), metric spaces with perfect projections.

Theorem 5. If a paracompact \(X\) of dimension \(\dim X=n\) admits a perfect mapping \(f:X\to R\) onto some metric space \(R\), then the space \(X\) is the limit of some spectrum
\[ X=\lim_{\leftarrow}\{R_\alpha,\widetilde{\omega}_\alpha^\beta\} \]
of \(n\)-dimensional in the sense of \(\dim\) metric spaces \(R_\alpha\), and all projections \(\widetilde{\omega}_\alpha^\beta\) are perfect mappings.

From this theorem it follows at once

Theorem 6. A Čech-complete paracompact space \(X\) of dimension \(\dim X=n\) is the limit of a certain spectrum of \(n\)-dimensional, in the sense of \(\dim\), complete metric spaces with perfect projections.

Proof of Theorem 5. Let \(\{\omega_\alpha\}_{\alpha\in\Omega}\) be the set of all covers of the space \(X\). To each cover \(\omega_\alpha\) we assign an \(n\)-dimensional, in the sense of \(\dim\), metric space \(R_\alpha\) and a perfect \(\omega_\alpha\)-mapping \(\mathfrak{F}_\alpha:X\to R_\alpha\). For every pair \(\alpha_1,\alpha_2\) there exists a perfect mapping
\[ \overline{\mathfrak{F}}_{(\alpha_1\alpha_2)}:X\to R_{\alpha_1}\times R_{\alpha_2}, \]
defined as follows:
\[ \overline{\mathfrak{F}}_{(\alpha_1\alpha_2)}(x)=\bigl(\mathfrak{F}_{\alpha_1}(x),\mathfrak{F}_{\alpha_2}(x)\bigr) \]
for all \(x\in X\). By Theorem 1 there exists an \(n\)-dimensional, in the sense of \(\dim\), metric space \(R_{(\alpha_1\alpha_2)}\) and perfect mappings
\[ \mathfrak{F}_{(\alpha_1\alpha_2)}:X\to R_{(\alpha_1\alpha_2)} \]
and
\[ h_{(\alpha_1\alpha_2)}:R_{(\alpha_1\alpha_2)}\to R_{\alpha_1}\times R_{\alpha_2}, \]
such that
\[ \overline{\mathfrak{F}}_{(\alpha_1\alpha_2)} = h_{(\alpha_1\alpha_2)}\mathfrak{F}_{(\alpha_1\alpha_2)}. \]

Let us now denote by \(p_{\alpha_i}\) the natural projections onto the factor spaces \(R_{\alpha_i}\), and put
\[ \mathfrak{F}_{\alpha_1}^{(\alpha_1\alpha_2)} = p_{\alpha_1}h_{(\alpha_1\alpha_2)},\qquad \mathfrak{F}_{\alpha_2}^{(\alpha_1\alpha_2)} = p_{\alpha_2}h_{(\alpha_1\alpha_2)}. \]
Obviously,
\[ p_{\alpha_1}\overline{\mathfrak{F}}_{(\alpha_1\alpha_2)} = \mathfrak{F}_{\alpha_1} = \mathfrak{F}_{\alpha_1}^{(\alpha_1\alpha_2)}\mathfrak{F}_{(\alpha_1\alpha_2)}, \qquad p_{\alpha_2}\overline{\mathfrak{F}}_{(\alpha_1\alpha_2)} = \mathfrak{F}_{\alpha_2} = \mathfrak{F}_{\alpha_2}^{(\alpha_1\alpha_2)}\mathfrak{F}_{(\alpha_1\alpha_2)}. \]
Next take, for each pair \((\alpha_1\alpha_2)\), \((\alpha_3\alpha_4)\), the product \(R_{(\alpha_1\alpha_2)}\times R_{(\alpha_3\alpha_4)}\) and repeat the preceding arguments. Continuing this process, we obtain a spectrum
\[ S=\{R_\alpha,\mathfrak{F}_\alpha^\beta\}, \]
where the indices \(\alpha\) denote collections \((\alpha_1\ldots \alpha_{2n})\). Repeating the arguments of Theorem 3, we conclude that
\[ X=\varprojlim \{R_\alpha,\mathfrak{F}_\alpha^\beta\}. \]
The theorem is proved.

I express my deep gratitude to V. Ponomarev and B. Pasynkov for the assistance rendered to me.

Moscow State University
named after M. V. Lomonosov

Received
29 V 1964

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