Abstract
Full Text
UDC 519.50+519.54
MATHEMATICS
M. M. CHOBAN
ON OPEN FINITE-TO-ONE MAPPINGS *
(Presented by Academician P. S. Aleksandrov, 17 VI 1966)
This work is closely connected with the recent works of A. V. Arkhangel’skii ((^3)) and V. V. Proizvolov ((^6)). Its main results are the following theorems.
Theorem 1. Let (f: X \to X) be an open finite-to-one mapping of a regular (p)-space ** (X) onto a metrizable space (Y). Then (X) is metrizable.
Theorem 2. The inverse image and the image of a hereditarily weakly paracompact space under open finite-to-one mappings are hereditarily weakly paracompact spaces (in the class of (T_2)-spaces).
Theorem 3. Let (f: X \to Y) be an open finite-to-one mapping of a regular space (X) onto a Hausdorff hereditarily paracompact space (Y). Then (X) too is hereditarily paracompact.
Remark. For the proof of Theorem 2 (3) it is sufficient to prove that the space (X) is weakly paracompact (paracompact), since every open subspace (U \subseteq X) is mapped openly and finite-to-one onto (fU).
We shall say that a set (A \subseteq X) is weakly paracompact (paracompact) in (X) if into every cover of the set (A) open in (X) one can inscribe some point-finite (locally finite) cover in (X) of the set (A) by sets open in (X).
Let (f: X \to Y) be a finite-to-one mapping. By (M_n) we shall agree to denote the totality of all points of the space (X) of multiplicity (n) with respect to the mapping (f) (i.e. (M_n=\mathscr E{x: f^{-1}fx) consists of (n) points(})).
Lemma 1. Let (f: X \to Y) be an open finite-to-one mapping of a Hausdorff space (X) onto a space (Y). If the set (fM_n) is weakly paracompact in (Y), then the set (M_n) is weakly paracompact in (X).
Proof. Let (\omega={U_\alpha}) be some cover, open in (X), of the set (M_n). Since the mapping (f) is open and finite-to-one, for any (n) points (x_1,x_2,\ldots,x_n \in M_n), for which (fx_1=fx_2=\cdots=fx_n), one can construct such open sets (Vx_1,Vx_2,\ldots,Vx_n) that (Vx_i \cap Vx_j=\varnothing) for (i \ne j), (fVx_1=fVx_2=\cdots=fVx_n), and each set (Vx_i) ((i=1,2,\ldots,n)) is contained in some set (U_\alpha \in \omega). Denote by (\eta={Vx}) the totality of the sets constructed by us, which, by construction, is inscribed in (\omega) and covers the set (M_n). Into the cover (f\eta={fVx}), open in (Y), of the set (fM_n), inscribe an open point-finite in (Y) cover (\xi={G_\beta \mid \beta \in \theta}) of the set (fM_n).
The system of sets open in (X), (f^{-1}\xi={f^{-1}G_\beta \mid \beta \in \theta}), is point-finite and covers (M_n). Each set (G_\beta) is contained in some (fVx); hence there exist such sets (Vx_1,Vx_2,\ldots,Vx_n \in \eta) that
(fVx_1=fVx_2=\cdots=fVx_n=fVx). Put (\Gamma_{\beta i}=f^{-1}G_\beta \cap Vx_i) and prove
* The mapping (f: X \to Y) is finite-to-one if the inverse image of every point consists of a finite number of points. Let us note that all mappings are assumed to be continuous and onto.
** The class of (p)-spaces was introduced and studied in the paper ((^4)).
the following relation:
[
\bigcup_{i=1}^{n} \Gamma_{\beta i}\cap M_n
=
f^{-1}G_\beta\cap M_n
\tag{a}
]
It is enough to verify that
[
\bigcup_{i=1}^{n} \Gamma_{\beta i}\cap M_n
\supset
f^{-1}G_\beta\cap M_n,
]
and the latter follows from the obvious relation
[
\bigcup_{i=1}^{n} Vx_i\cap M_n
=
f^{-1}fVx\cap M_n.
]
For each set (G_\beta) we fix some (fVx\supset G_\beta) and put
(\gamma={\Gamma_{\beta i}\mid \beta\in\theta;\ i=1,2,\ldots,n}). On the basis of formula (a), the system (\gamma) covers the set (M_n), and, by construction, (\gamma) is inscribed in (\omega). Let now (x) be an arbitrary point of the space (X). By the hypothesis, the point (x) is contained in no more than finitely many elements of the system (f^{-1}\xi)—say, in (f^{-1}G_{\beta_1}, f^{-1}G_{\beta_2},\ldots,f^{-1}G_{\beta_k}). Then (x) can belong only to the sets (\Gamma_{\beta_l i}), where (l=1,\ldots,k) and (i=1,2,\ldots,n). Therefore the system (\gamma) is point-finite in (X). Lemma 1 is proved.
Similarly one proves
Lemma 2. Let (f:X\to Y) be an open finite-to-one mapping of a Hausdorff space (X) onto a space (Y). If the set (fM_n) is paracompact in the subspace
[
Y_n=Y\setminus\bigcup_{i=1}^{n-1} fM_i
\quad (\text{where } Y_1=Y),
]
then the set (M_n) is paracompact in the subspace
[
X_n=X\setminus\bigcup_{i=1}^{n-1} M_i
\quad (\text{where } X_1=X).
]
Proof of Theorem 2. If the space (X) is hereditarily weakly paracompact, then it is not hard to see that the space (Y) is also hereditarily weakly paracompact. Let now the space (Y) be hereditarily weakly paracompact and let (\omega={U_\alpha\mid \alpha\in\theta}) be some open cover of the space (X).
From Theorem 1 of [2] it follows that, for any natural number (n), the set
[
\bigcup_{i=1}^{n} M_i
]
is closed in (X). In view of the hereditary weak paracompactness of the space (Y), the sets (M_n) and (fM_n), for any natural number (n), satisfy Lemma 1; consequently, one can find an open system (\omega_n), point-finite in (X), which covers the set (M_n) and is inscribed in the system
[
\omega_n'=\left{\,U_\alpha\cap\left(\bigcup_{j=n}^{\infty} M_j\right)\mid \alpha\in\Theta\,\right}.
]
It is clear that the cover
[
\omega'=\bigcup_{n=1}^{\infty}\omega_n
]
of the space (X) is point-finite. This proves Theorem 2.
Theorem 4. The preimage and the image of a hereditarily finally compact space under open finite-to-one mappings is again a hereditarily finally compact space (in the class of Hausdorff spaces).
The proof of this fact is analogous to the proof of Theorem 2; only, instead of point-finite systems, one must consider countable systems.
Proof of Theorem 3. Let (Y) be a hereditarily paracompact space. By Lemma 2, the set (M_1) is paracompact in (X). Suppose that we have already proved that the set
[
\bigcup_{i=1}^{n} M_i
]
is paracompact in (X). Consider an arbitrary cover (\omega) of the set
[
\bigcup_{i=1}^{n+1} M_i.
]
By the hypothesis, in (\omega) one can inscribe some open locally finite cover
[
\omega_1={U_\alpha\mid \alpha\in A}
]
of the set
[
\bigcup_{i=1}^{n} M_i,
]
and, by Lemma 2, also some open ...
a locally finite in (X\setminus \bigcup_{i=1}^{n} M_i) cover (\omega_2={V_\beta\mid \beta\in B}) of the set (M_{n+1}). Since (X) is regular and, by assumption, the set (\bigcup_{i=1}^{n} M_i) is paracompact in (X), there exists an open set (\Gamma\supset M_{n+1}\setminus G) in (X), where (G=\bigcup_{\alpha\in A} U_\alpha), such that
[
[\Gamma]\cap\left(\bigcup_{i=1}^{n} M_i\right)=\varnothing.
]
Therefore the system (\omega_2'={V_\beta\cap\Gamma\mid \beta\in B}) is locally finite in (X), and, consequently, the cover (\omega'=\omega_1\cup\omega_2') is locally finite in (X).
Thus, the set (\bigcup_{i=1}^{n} M_i) is paracompact in (X) for every natural number (n); consequently, it can be separated from any closed set in (X) that does not meet it. Let now (\gamma) be an arbitrary open cover of the space (X). For every natural number (n), by Lemma 2, there exists a certain (\sigma)-discrete in
[
X_n=X\setminus \bigcup_{i=1}^{n-1} M_i\quad (X_1=X)
]
cover
[
\gamma_n={U_{\alpha n}\mid \alpha\in A}
]
of the set (M_n), inscribed in (\gamma). Suppose that (\sigma)-discrete in (X) systems
[
\gamma_1'={\Gamma_{\alpha1}\mid \alpha\in A},\ldots,\gamma_n'={\Gamma_{\alpha n}\mid \alpha\in A},
]
have been constructed, covering the set (\bigcup_{i=1}^{n} M_i), and moreover (\gamma_1'=\gamma_1). The system (\gamma_{n+1}'), (\sigma)-discrete in (X), is constructed as follows: put
[
G=\bigcup_{i=1}^{n}\left(\bigcup_{\alpha\in A}\Gamma_{\alpha i}\right)
]
and take such an open in (X_n) set (\Gamma) that
[
\Gamma\supset M_{n+1}\setminus G
\quad\text{and}\quad
[\Gamma]\cap\left(\bigcup_{i=1}^{n} M_i\right)=\varnothing.
]
The system
[
\gamma_{n+1}'={\Gamma_{\alpha(n+1)}=\Gamma\cap U_{\alpha(n+1)}\mid \alpha\in A}
]
is the desired one.
Thus, into the cover (\gamma) we have inscribed the (\sigma)-discrete cover
[
\gamma'=\bigcup_{n=1}^{\infty}\gamma_n'.
]
On the basis of Proposition 1 from (5), the space (X) is paracompact.
Proposition 1. If a Hausdorff topological space (X) is mapped openly and finite-to-one onto a perfectly normal paracompact space (Y), then into every open cover (\omega) of the space (X) one can inscribe an open (\sigma)-discrete cover (\gamma).
The proof of this fact is analogous to the proof of Theorem 3. Theorem 1 follows from Theorem 3 and from a theorem of A. V. Arhangel’skii from (3).
Example 1. The points of the space (R) are all points of the segment ([0,1]). At all points (x\ne0), the neighborhoods are the same as on the half-interval ((0,1]) (see (1), p. 863). As neighborhoods of the point (x=0) take sets of the form ([0,\varepsilon)\setminus D), where (D={1,1/2,\ldots,1/n,\ldots}) and (0<\varepsilon<1). The space (R) is not paracompact (it is not regular), although it is finally compact. Take (I=[0,1]) in the usual topology. Put (Z=R\cup I), where ([R]_Z\cap[I]_Z=\varnothing). Glue the points (0\in R) and (0\in I) into one; endow the resulting set with the quotient topology and denote it by (X). The space (X): 1) is Hausdorff; 2) is not paracompact; 3) is mapped openly and finite-to-one onto the segment (I=[0,1]); 4) is a (p)-space (as the sum of a countable number of bicompacts of countable character in (X)). This example shows that Theorems 1 and 3 cannot be extended from regular spaces to Hausdorff ones.
Example 2. Let (X) be the well-known Niemytzki space. It is not difficult to verify that (X) has a countable refining sequence of covers; hence it is a (p)-space. Denote by (L_0\subseteq X) the set of real numbers, and by (L_n\subseteq X) a line parallel-
the line (L_0), which is at distance (1/n) from (L_0). Put
[
Z=\bigcup_{i=0}^{\infty} L_i.
]
The space (Z) is a subspace of the space (X). The space (Z): 1) is completely regular; 2) is a (p)-space; 3) is not weakly paracompact; 4) is mapped openly, countably, and compactly onto the set of real numbers (L_0) in the usual topology (it is enough to map points of the form ((x,0)), ((x,1/n)) to the point (x)). This example shows that the preceding theorems cannot be extended to countable-to-one mappings. Example 1 from (6) shows that the word “hereditarily” in Theorems 2, 3, and 4 cannot be omitted. There exists an example (see (6), Example 2) of a finite-to-one open mapping of a perfectly normal space with the first axiom of countability onto a compact space; therefore the requirement of “featheredness” in the formulation of Theorem 1 is essential.
Moscow State University
named after M. V. Lomonosov
Received
7 VI 1966
REFERENCES
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