MATHEMATICS

A. LELEK

ON AN ESTIMATE OF THE DEFECT OF A SPACE

(Presented by Academician P. S. Aleksandrov, 12 I 1967)

Denote by \(\operatorname{def} X\) the defect of the space \(X\), and by \(P(X)\) the set of all points of the space \(X\) at which it is peripherally compact. There are \((^1)\)** examples of spaces \(X\) such that the set \(X \setminus P(X)\) is compact and the defect \(\operatorname{def} X\) is equal to \(\dim (X \setminus P(X)) + 1\). We shall prove a theorem from which the inequality

\[ \operatorname{def} X \leq \dim (X \setminus P(X)) + 1 \]

follows, under the condition that the set \(P(X)\) is open. All spaces considered will be metric spaces with a countable base.

Lemma 1. If \(X\) is a totally bounded space and \(\dim X \leq n\), then there exists a sequence \(\varphi_0, \varphi_1, \ldots\) of closed finite covers of the space \(X\) such that \(\varphi_0=\{X\}\), the diameter of each element of the cover \(\varphi_i\) is less than \(i^{-1}\), and the multiplicity of the cover \(\varphi_{i-1}\cup \varphi_i\) is less than \(n+3\) \((i=1,2,\ldots)\).

Proof. We shall construct the covers \(\varphi_i\) by induction with respect to \(i\). The cover \(\varphi_0\) is defined and satisfies the following condition \((\mathfrak M_i)\) for \(i=0\):

\((\mathfrak M_i)\). The dimension of the common part of any family consisting of \(r\leq n+2\) elements of the cover \(\varphi_i\) is less than \(n-r+2\).

Suppose now that \(i>0\) and that a closed finite cover \(\varphi_{i-1}=\{F_1,\ldots,F_l\}\) of the space \(X\) has been defined, satisfying condition \((\mathfrak M_{i-1})\). Decompose the space \(X\) into the sum of open sets \(G_1,\ldots,G_m\), the diameter of each of which is less than \(i^{-1}\). By Menger’s theorem \((^2)\), there exists a closed cover \(\varphi_i=\{F'_1,\ldots,F'_m\}\) of the space \(X\) such that \(F'_k\subseteq G_k\) for \(k=1,\ldots,m\), and such that condition \((\mathfrak M_i)\) is satisfied together with the inequality

\[ \dim (F_{j_0}\cap \cdots \cap F_{j_r}\cap F'_{k_0}\cap \cdots \cap F'_{k_s}) < \dim (F_{j_0}\cap \cdots \cap F_{j_r})-s \]

for any integers \(s\) in the interval \(0\leq s\leq d+1\), where

\[ d=\dim (F_{j_0}\cap \cdots \cap F_{j_r}); \]

* The least dimension of a remainder in compact extensions of a given space is called the defect of this space.

* A space \(X\) is called peripherally compact at a point* \(x\in X\) if \(x\) has arbitrarily small neighborhoods in \(X\) whose boundaries are compact.

*** The present paper to some extent supplements paper \((^1)\), two assertions of which (namely, Corollaries 2.1 and 2.2) are not fully justified. We note that, in order to leave the assertion in Theorem 2 from \((^1)\) unchanged, one must change the definition of the quantity \(\operatorname{Com} X\), which is given in \((^1)\). One possible approach to this problem is to change only the first step of the inductive definition of the inequality \(\operatorname{Com} X\leq n\), with the induction beginning not from \(n=-1\), but from \(n=0\). Thus, following Grot, the relation \(\operatorname{Com} X\leq 0\) is defined as a condition equivalent to the equality \(X=P(X)\). However, then the inequality \(\operatorname{def} X\leq \operatorname{Com} X\) does not follow from the work of Vries. The theorem given below entails Corollaries 2.1 and 2.2 from \((^1)\) for those spaces \(X\) for which the set \(P(X)\) is open.

\(0 \leqslant j_0 < \cdots < j_r \leqslant l,\quad 0 \leqslant k_0 < \cdots < k_s \leqslant m.\) According to \((\mathfrak R_{i-1})\), we have \(d \leqslant n-r\), if \(r \leqslant n+1\). Take integers \(r \geqslant 0,\ t \geqslant 0\) such that \(r+t=n+1\). Defining the number \(s\) by the formula

\[ s= \begin{cases} t, & \text{if } t \leqslant d+1,\\ d+1, & \text{if } t>d+1, \end{cases} \]

we obtain \(0 \leqslant s \leqslant d+1,\ s \leqslant t\). Moreover, \(d-t \leqslant n-r-t=-1\). Then the common part of any family consisting of \(r+1\) elements of the cover \(\varphi_{i-1}\) and \(t+1\) elements of the cover \(\varphi_i\) has dimension

\[ \dim(F_{j_0}\cap\cdots\cap F_{j_r}\cap F'_{k_0}\cap\cdots\cap F'_{k_t})\leqslant \]

\[ \leqslant \dim(F_{j_0}\cap\cdots\cap F_{j_r}\cap F'_{k_0}\cap\cdots\cap F'_{k_s}) \leqslant d-s= \begin{cases} d-t=-1,\\ d-(d+1)=-1. \end{cases} \]

Thus, the multiplicity of the cover \(\varphi_{i-1}\cup\varphi_i\) is less than \(n+3\); Lemma 1 is proved.

Let \(\alpha,\beta\) be some families of sets. We shall denote by \(\alpha\wedge\beta\) the family of all sets of the form \(A\cap B\), where \(A\in\alpha,\ B\in\beta\). The closure of a set \(A\) is denoted by \(\operatorname{cl} A\). If \(\alpha\) is a family of sets, then \(\operatorname{cl}\alpha\) denotes the family of all sets \(\operatorname{cl} A\), where \(A\in\alpha\). Further, \(|\alpha|\) is the union of all sets of the family \(\alpha\). Finally, let \(I^{\aleph_0}_0\) denote the first left face of the Hilbert cube \(I^{\aleph_0}\), i.e.,

\[ I^{\aleph_0}_0=\{0\}\times[0,1]\times[0,1]\times\cdots \subset[0,1]\times[0,1]\times[0,1]\times\cdots=I^{\aleph_0}. \]

Lemma 2. If \(X\) is a closed subset of a space \(Y\), then there exists a continuous mapping \(f:Y\to I^{\aleph_0}\) such that \(f|X\) is a homeomorphism and

\[ f(X)\subseteq I^{\aleph_0}_0,\qquad f(Y\setminus X)\subseteq I^{\aleph_0}\setminus I^{\aleph_0}_0,\qquad \dim f(Y)\leqslant \dim X+1. \]

Proof. We may immediately assume that \(X\ne\varnothing\) and \(Y\subseteq I^{\aleph_0}\). The mapping \(f\) will be an extension of the embedding \(h:X\to I^{\aleph_0},\ h(y)=y\) \((y\in X)\). This extension is trivial in the case when the set \(X\) is finite-dimensional. Thus, putting \(n=\dim X\), consider a sequence of closed finite covers \(\varphi_0,\varphi_1,\ldots\) of the set \(X\) such that the assertion of Lemma 1 is satisfied. We may assume that all elements of the families \(\varphi_i\) are nonempty. Take \(\gamma_{-1}=\{\varnothing\},\ \gamma_0=\{Y\}\), and note that the family \(\gamma_0\) satisfies the following condition \((\mathfrak R_i)\) for \(i=0\):

\((\mathfrak R_i)\) The multiplicity of both the family \(\operatorname{cl}\gamma_{i-1}\cup\operatorname{cl}\gamma_i\) and the family \(\operatorname{cl}\gamma_i\cup\varphi_{i+1}\) is less than \(n+3\).

Suppose that for some \(i>0\) a family \(\gamma_{i-1}\) has been defined such that \(X\subseteq|\gamma_{i-1}|\), \(\gamma_{i-1}\) consists of open subsets of the space \(Y\), and satisfies condition \((\mathfrak R_{i-1})\). We construct the family \(\gamma_i\). Let \(F\in\varphi_i\). Let \(F'\) be the union of all sets each of which is a common part, intersecting \(F\), of a family of sets contained in the family \(\operatorname{cl}\gamma_{i-1}\cup\varphi_i\) or in the family \(\varphi_i\cup\varphi_{i+1}\). Since \(F\subseteq|\gamma_{i-1}|\setminus F'\) and the diameter of \(F\) is less than \(i^{-1}\), there exists an open subset \(G\) of the space \(Y\) such that \(F\subseteq G\subseteq\operatorname{cl}G\subseteq|\gamma_{i-1}|\setminus F'\) and the diameter of \(G\) is less than \(i^{-1}\). But from condition \((\mathfrak R_{i-1})\) and Lemma 1 it follows that the multiplicity of both families \(\operatorname{cl}\gamma_{i-1}\cup\varphi_i,\ \varphi_i\cup\varphi_{i+1}\) is less than \(n+3\). Consequently, less than \(n+3\) also are the multiplicities of the families \(\operatorname{cl}\gamma_{i-1}\cup\operatorname{cl}\varphi'_i,\ \operatorname{cl}\varphi'_i\cup\varphi_{i+1}\), where \(\varphi'_i\) is the family obtained from \(\varphi_i\) by replacing the set \(F\) by the set \(G\). Replacing successively all sets of the family \(\varphi_i\), we thus obtain a family \(\gamma_i\) such that \(X\subseteq|\gamma_i|\subseteq\operatorname{cl}|\gamma_i|\subseteq|\gamma_{i-1}|\), \(\gamma_i\) consists of open subsets of the space \(Y\) and satisfies condition \((\mathfrak R_i)\). Moreover, the elements of the family \(\gamma_i\) intersect \(X\) and their diameters are less than \(i^{-1}\).

It follows from this that \(Y \setminus X\) is the sum of the sets \(|\gamma_i| \setminus |\gamma_{i+1}|\) \((i=0,1,\ldots)\). Put
\[ \chi_i=\gamma_i \wedge \{\,Y\setminus \operatorname{cl}|\gamma_{i+2}|\,\}. \]
We have
\[ |\gamma_i|\setminus |\gamma_{i+1}|\subset |\gamma_i|\setminus \operatorname{cl}|\gamma_{i+2}|=|\chi_i|\subset \operatorname{cl}|\chi_i|\subset Y\setminus |\gamma_{i+2}|\subset Y\setminus X. \]
Thus, \(\chi=\chi_0\cup\chi_1\cup\cdots\) is an open countable covering of the set \(Y\setminus X\); the closures of the elements of the covering \(\chi\) lie in \(Y\setminus X\), and their diameters tend to zero. Moreover, in the family \(\chi\) each element intersects only a finite number of elements, and from condition \((\mathfrak{M}_i)\) it follows that the multiplicity of \(\chi\) is less than \(n+3\). Applying Kuratowski’s method \((^3)\) to the system \(\chi\) and to its nerve, we obtain the required extension \(f\) of the embedding \(h\). The set \(f(Y\setminus X)\) is contained in the nerve of the covering \(\chi\), which is an infinite polyhedron lying in \(I^{\aleph_0}/I^{\aleph_0}_0\) and having dimension less than \(n+2\); Lemma 2 is proved.

Theorem. For every space \(X\) the inequality
\[ \operatorname{def} X \leq \dim \operatorname{cl}(X\setminus P(X)) + 1 \]
holds.

Proof. Denote by \(A\) the closure of the set \(X\setminus P(X)\). Let \(n=\dim A\). Take such a compact extension \(cX\) of the space \(X\) that the closure \(cA\) of the set \(A\) in \(cX\) has dimension \(n\). From Lemma 2 there follows the existence of a continuous mapping
\[ f:cX\to I^{\aleph_0}_0 \]
such that \(f|cA\) is a homeomorphism, \(f(cA)\cap f(cX\setminus cA)=\varnothing\), and \(\dim f(cX)\leq n+1\). There is a sequence of open finite coverings \(\gamma_1,\gamma_2,\ldots\) of the image \(f(cX)\), possessing the basis property at each of its points and such that \(\gamma_{i+1}\) is star-refined into \(\gamma_i\), and the multiplicity of \(\gamma_i\) is less than \(n+3\) \((i=1,2,\ldots)\). Consider the open finite covering \(\chi_i\) of the space \(X\), consisting of the sets \(f^{-1}(G)\cap X\), where \(G\in\gamma_i\). Then the sequence of coverings \(\chi_1,\chi_2,\ldots\) has the basis property at the points of the set \(A\). Since \(X\setminus A\subset P(X)\), there exists such a sequence \(K_1,K_2,\ldots\) of open subsets of the space \(X\) that all boundaries \(\operatorname{cl} K_i\setminus K_i\) are compact and every point of the set \(X\setminus A\) lies in sets \(K_i\) of sufficiently small diameter. Put
\[ \varkappa_i=\{K_i,\ X\setminus \operatorname{cl} K_i\},\qquad \lambda_j=\chi_j\wedge\varkappa_1\wedge\varkappa_2\wedge\cdots\wedge\varkappa_j \quad (j=1,2,\ldots). \]
The set \(X\setminus |\lambda_j|\) is the sum of the boundaries \(\operatorname{cl} K_i\setminus K_i\), \(i=1,\ldots,j\). Hence all \(\lambda_j\) are borderings* of the space \(X\), and the sequence \(\lambda_1,\lambda_2,\ldots\) has the basis property at all points of this space. On the other hand, the bordering \(\lambda_{j+1}\) is star-refined into the bordering \(\lambda_j\); the multiplicity of \(\lambda_j\) is less than \(n+3\) \((j=1,2,\ldots)\). According to a theorem of Yu. M. Smirnov \((^4)\), the inequality
\[ \operatorname{def} X\leq n+1 \]
is obtained.

Mathematical Institute
Polish Academy of Sciences

Received
9 I 1967

REFERENCES

1. A. Lelek, DAN, 160, No. 3, 534 (1965).
2. K. Menger, Dimensionstheorie, Leipzig—Berlin, 1928, S. 170.
3. C. Kuratowski, Topologie, 1, Warszawa, 1952, p. 215–217.
4. Yu. M. Smirnov, DAN, 168, No. 3, 528 (1966).

* A sequence of families \(\gamma_1,\gamma_2,\ldots\) has the basis property at a point \(x\in X\) if for every neighborhood \(U\subseteq X\) of the point \(x\) there exist an integer \(i>0\) and a neighborhood \(V\subseteq X\) of the point \(x\) such that from the conditions \(G\in\gamma_i,\ G\cap V\ne\varnothing\) it follows that \(G\subset U\).

* A finite family \(\gamma\), consisting of open subsets of the space \(X\), is called a bordering* \((^4)\) if \(X\setminus |\gamma|\) is compact.