UDC 513.83

MATHEMATICS

L. G. ZAMBACHIDZE

ON SOME PROPERTIES OF SPACES WITH BICOMPACT REMAINDERS OF FINITE ORDERS

(Presented by Academician P. S. Aleksandrov on 31 VII 1969)

1. In this paper we study some properties of spaces of Iséki orders \(n\), i.e. of the classes \(K_n\) \((n \geqslant -1)\), and of their mappings. The spaces are assumed to be completely regular and Hausdorff, and the mappings continuous.

As is known \((^1)\), a space \(X\) belongs to the class \(K_n\) \((n \geqslant 0)\) in the sense of Iséki if and only if \(R^{n/2}(X)\) is bicompact when \(n\) is even, and \(R^{(n-1)/2}(X)\) is locally bicompact when \(n\) is odd. \(R^i(X)\) is defined as follows: \(R^0(X)=X\), \(R^1(X)\) is the set of all points of non-local bicompactness of \(X\), and \(R^i(X)=R^1[R^{i-1}(X)]\) for \(i>1\).

In \((^1)\) an “external” characteristic of the spaces of the classes \(K_n\) is also given: a space \(X\) belongs to the class \(K_n\) if and only if the remainder of order \(n\) \((^1)\) is bicompact. It is clear that \(K_0\) coincides with the class of bicompacts, and \(K_1\) with the class of locally bicompact spaces. We shall further assume that the class \(K_{-1}\) consists of empty spaces.

In what follows another characteristic of the classes \(K_n\) will be needed.

Theorem 1. A space \(X\) belongs to the class \(K_n\) \((n \geqslant 1)\) if and only if \(R^1(X)\) belongs to the class \(K_{n-2}\).

It is shown in \((^1)\) that a closed subset of a space from the class \(K_n\) also belongs to the class \(K_n\). For open subsets the following holds.

Theorem 2. Let \(X\) be a space of the class \(K_n\), and let \(G \subseteq X\) be an open subset of it. Then \(G\) belongs to the class \(K_{n+1}\) if \(n\) is even, and to the class \(K_n\) if \(n\) is odd.

It is known that local bicompactness is preserved under open mappings. For spaces from the classes \(K_n\) the following holds.

Theorem 3. The open image of a space of the class \(K_n\) belongs to the same class \(K_n\).

In \((^1)\) the question of the topological product of spaces from the classes \(K_n\) was considered in the special case when one of the factors is either bicompact or locally bicompact. In the general case the following holds.

Theorem 4. Let \(X\) belong to the class \(K_m\), and \(Y\) to the class \(K_n\); then \(X \times Y\) belongs to the class \(K_{m+n}\) if either \(m\) or \(n\) is even, and to the class \(K_{m+n-1}\) if \(m\) and \(n\) are odd.

The next theorem gives an internal characteristic of a space with a bicompact remainder of finite order, i.e. of some class \(K_n\), and is an answer to a question posed by G. S. Chogoshvili.

Theorem 5. A space \(X\) has a bicompact remainder of finite order if and only if it can be decomposed into the sum of a finite number of pairwise disjoint subsets, each of which is locally bicompact in the induced topology.

For closed mappings of spaces from the classes the following theorems hold:

Theorem 6. Let \(X\) be a hereditarily weakly paracompact space and let \(f:X \to Y\) be a closed mapping. Then, if \(X\) belongs

class \(K_n\), then \(Y\) belongs to the class \(K_{2n}\) for even \(n\) and to the class \(K_{2n+1}\) for odd \(n\).

Theorem 7. Let \(X\) be a hereditarily weakly paracompact space and \(f:X\to Y\) a closed mapping such that \(R(f)\) (see (1)) is bicompact. Then, if \(X\) belongs to the class \(K_n\), then \(Y\) belongs to the class \(K_n\) for even \(n\) and to the class \(K_{n+1}\) for odd \(n\).

The following theorem is a strengthening of a theorem of Lelek from \((^2)\).

Theorem 8. Let \(X\) be a finally compact space of countable type and let \(f:X\to Y\) be a mapping such that \(f^{-1}(y)\) belongs to \(K_n\) for every \(y\in Y\). Then
\[ \dim X \leq \dim f(X)+\max\{\dim f,\operatorname{def}X\}. \]

Definition 1. A space \(X\) is called \(n\)-peripheral if in this space there exists a base of open sets whose boundaries belong to the class \(K_n\).

The following theorem may be regarded as a generalization of the Freudenthal–Morita theorem \((^3)\), obtained for \(n=-1\).

Theorem 9. Let \(X\) be a space having a base of open sets \(U=\{U_\alpha\}_{\alpha\in M}\) with the following properties: 1) the boundary \(\operatorname{Fr}U_\alpha\) belongs to the class \(K_n\) for every \(\alpha\in M\); 2) if \(U_\alpha\in U\), then \(X\setminus [U_\alpha]\in U\); 3) for every closed \(A\subseteq U_\alpha\in U\) there exists \(U_\beta\in U\) such that \(A\subseteq [U_\beta]\subseteq U_\alpha\); 4) if \(U_\alpha\in U\) and \(U_\beta\in U\), then \(U_\alpha\cup U_\beta\in U\). Then there exists a bicompact extension \(bX\) with an \((n-1)\)-peripheral remainder \(bX\setminus X\).

Definition 2. A base \(U=\{U_\alpha\}_{\alpha\in M}\) of open sets of \(X\) is called regular if the following conditions are satisfied: 1) if \(U_\alpha\in U\) and \(U_\beta\in U\), then \(U_\alpha\cup U_\beta\in U\); 2) if \(U_\alpha\in U\), then \(X\setminus [U_\alpha]\in U\); 3) if \(U_\alpha\in U\) and \(U_\beta\in U\), then \(U_\alpha\setminus [U_\beta]\in U\).

If the Freudenthal–Morita theorem gives conditions under which there exists an extension with a zero-dimensional first remainder, then the following theorems give conditions for the existence of extensions with zero-dimensional remainders of the 2nd and 3rd orders. Zero-dimensionality is understood in the sense of ind.

Theorem 10. Let \(X\) be a paracompact \(1\)-peripheral space. Then there exists an extension of order 2 with a zero-dimensional remainder.

Theorem 11. Let \(X\) be a paracompact space of countable type and let \(U=\{U_\alpha\}_{\alpha\in M}\) be the collection of all open subsets of \(X\) whose boundaries are bicompactly situated in the sense of Chogoshvili \((^4)\) and belong to the class \(K_2\). If \(U\) is a regular base, then there exists an extension of order 3 with a zero-dimensional remainder.

The following theorem gives a condition under which spaces of the classes \(K_n\) are complete in the sense of Čech.

Theorem 12. Let \(X\) be a completely normal space of class \(K_n\) which is completely normally embedded \((^5)\) in some bicompact extension. Then \(X\) is complete in the sense of Čech.

Corollary. A metrizable space of class \(K_n\) is an absolute \(G_\delta\).

Remark 1. For arbitrary completely normal spaces Theorem 12 is not true; there even exists a countable completely normal space of class \(K_2\) which is not complete in the sense of Čech. However, the following holds.

Theorem 13. A countable space of class \(K_n\) is complete in the sense of Čech if and only if it is metrizable.

There exists a space of class \(K_n\) which is not a \(k\)-space. We give one sufficient condition under which a space of class \(K_n\) is a \(k\)-space.

Theorem 14. Let \(X\) be a space of class \(K_n\) such that for every bicompactum \(\Phi\) from \(R^1(X)\) there exists a bicompactum \(F\supseteq \Phi\) of countable character in \(X\). Then \(X\) is a \(k\)-space.

1. In the survey article \((^6)\) A. V. Arhangel'skii posed the following problems:

Problem 1. Let \(X\) be a finally compact space and \(f: X \to Y\) its closed mapping. Is the set
\[ \Phi=\{y: Y\ f^{-1}(y)\ \text{is not bicompact}\} \]
always at most countable?

Problem 2. Is every strict \(p\)-space perfectly mapped onto some space with a refining sequence of covers?

Below negative solutions of both of the indicated problems are given. Namely, in 2.1 an example will be constructed of a finally compact space of class \(K_2\) in the sense of Isasardze, and of such a closed mapping \(f: X \to Y\) of it that the set \(\Phi\) is open, everywhere dense in \(Y\), and has the cardinality of the continuum; in 2.2 an example will be constructed of a locally bicompact, weakly paracompact and \(\sigma\)-paracompact, nonnormal strict \(p\)-space which cannot be perfectly mapped onto any space with a refining sequence of covers. In constructing these examples one construction from \((^7)\) is used, applied there for another purpose.

2.1. Let \(OXYZ\) be some rectangular coordinate system in \(R^3\), and let \(K^3\) be the unit cube:
\[ K^3=E\{(x,y,z)\in R^3: 0\le x\le 1,\ 0\le y\le 1,\ 0\le z\le 1\}. \]
Consider the opposite faces \(\alpha\) and \(\beta\) of the cube \(K^3\):
\[ \alpha=E\{(x,y,z)\in K^3: 0\le x\le 1,\ y=0,\ 0\le z\le 1\} \]
and
\[ \beta=E\{(x,y,z)\in K^3: 0\le x\le 1,\ y=1,\ 0\le z\le 1\}. \]
Let
\[ \beta^*=E\{(x,y,z)\in\beta,\ z\ \text{rational}\}. \]
We introduce a topology on the set \(X=\alpha\cup\beta^*\) in the following way: if a point \(M\in\beta^*\), then it is isolated in \(X\); if, however, a point \(M\in\alpha\), then its neighborhood is
\[ OM=O'M\cup \{P(O'M)\setminus A\}, \]
where \(O'M\) is an open rectangular neighborhood of the point \(M\) in the usual topology of the face \(\alpha\); \(P(O'M)\) is the projection onto \(\beta^*\); and \(A\) is some countable subset of \(\beta^*\). It can be shown that \(X\) is regular, finally compact, and belongs to the class \(K_2\). Let
\[ \Phi_1=E\{(x,y,z)\in\alpha: 0\le x\le 1,\ y=0,\ z=0\} \]
and
\[ \Phi_2=E\{(x,y,z)\in\beta^*: 0\le x\le 1,\ y=1,\ z=0\}. \]
The set \(Y=\Phi_1\cup\Phi_2\subseteq X\) is closed in \(X\). Therefore \(Y=\Phi_1\cup\Phi_2\), in the induced topology, also has the listed properties of \(X\). The mapping \(f: X\to Y\) is defined as follows:
\[ f(x,y,z)=(x,1,0),\quad \text{if } M(x,y,z)\in\beta^*, \]
and
\[ f(x,y,z)=(x,0,0),\quad \text{if } M(x,y,z)\in\alpha. \]
It can be shown that the mapping \(f\) is closed, and the set \(\Phi_2\), consisting of the set of all points whose full inverse images are not bicompact, is open, everywhere dense in \(Y\), and has the cardinality of the continuum.

2.2. Let \(OXYZ\) be some rectangular coordinate system in \(R^3\), and let \(K^3\) be the unit cube:
\[ K^3=E\{(x,y,z)\in R^3: 0\le x\le 1,\ 0\le y\le 1,\ 0\le z\le 1\}. \]
Consider the sets \(\alpha\) and \(\beta\) of the cube \(K^3\), defined in the following way:
\[ \alpha=E\{(x,y,z)\in K^3: x\in C,\ y=0,\ z\in C\} \]
and
\[ \beta=E\{(x,y,z)\in K^3: x\in C,\ y=0,\ z\in C\}, \]
where \(C\) denotes some fixed zero-dimensional compact subset of the interval \([0,1]\) of the cardinality of the continuum (for example, the Cantor perfect set). Define the set \(\Phi\) as follows:
\[ \Phi=E\{(x,y,z)\in\alpha: x\in C,\ y=0,\ z=0\}. \]

On the set
\[ X=(\alpha\setminus\Phi)\cup\beta \]
we introduce a topology in the following way: represent \(\beta\) as the disjoint sum of the sets \(C_\lambda\), i.e.
\[ \beta\bigcup_\lambda C_\lambda,\quad C_\lambda\cap C_\gamma=\varnothing,\quad \text{for } \lambda\ne\gamma, \]
where each \(C_\lambda=C\) has the topology induced from the interval \([0,1]\), parallel to the axis \(OZ\). Let \(x\in\beta\); then there exists a unique \(C_{\lambda(x)}\) such that \(x\in C_{\lambda(x)}\). As a neighborhood of the point \(x\) we declare an arbitrary open neighborhood of it in the topology of \(C_{\lambda(x)}\). If the point \(x\in\alpha\setminus\Phi\), then its neighborhood \(Ox\) in \(X\) is defined by the formula
\[ Ox=O'x\cup \{P(O'x)\setminus B\},\quad O'x=Vx\cap(\alpha\setminus\Phi), \]
where \(Vx\) is a rectangular neighborhood of the point \(x\) in the natural topology of the face \(\alpha\), having compact closure in \(\alpha\); \(P(O'x)\) is the projection of \(O'x\) onto \(\beta\), and the set \(B\) is either empty or ...

\[ B=\bigcup_{i=1}^{n}\Phi_{\lambda_i}^{\prime}, \]
where each \(\Phi_{\lambda_i}^{\prime}=T_i\cap C_{\lambda_i}\), and \(T_i\) is an open connected set on the interval \([0,1]\), containing \(C_{\lambda_i}\).

It can be shown that the space \(X\) in this topology is locally bicompact, \(\sigma\)-paracompact, weakly paracompact, nonnormal, a strict \(p\)-space, and does not admit a perfect mapping onto a space with a refining sequence of covers.

Tbilisi State
University

Received
4 VI 1969

REFERENCES

\({}^{1}\) Kh. N. Inasaridze, DAN, 166, No. 5, 1043 (1966).
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\({}^{3}\) E. G. Sklyarenko, Izv. AN SSSR, ser. matem., 26, 427 (1962).
\({}^{4}\) G. S. Chogoshvili, Tr. Tbilissk. matem. inst., 18 (1950).
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\({}^{6}\) A. V. Arkhangel’skii, UMN, 21, issue 4 (1966).
\({}^{7}\) V. V. Proizvolov, Vestn. Mosk. univ., ser. matem., mekh., 13, 29 (1962).