MATHEMATICS

E. G. SKLYARENKO

ON PERFECT BICOMPACT EXTENSIONS

(Presented by Academician P. S. Aleksandrov, 14 X 1960)

Let \(X\) be a completely regular space. A bicompact extension \(Y\) of the space \(X\) is called perfect with respect to an open set \(U \subset X\) if the closure in \(Y\) of the boundary \(\operatorname{Fr}_X U\) of the set \(U\) in \(X\) is the boundary in \(Y\) of the set \(O\langle U\rangle^*\). The extension \(Y\) is called simply perfect if it is perfect with respect to every open set of the space \(X\).

Lemma 1. Let \(\delta\) be the proximity relation on the space \(X\) corresponding to the bicompact extension \(Y\). The bicompact extension \(Y\) of the space \(X\) is perfect with respect to the open set \(U \subset X\) if and only if, for every set \(A \subset U\), from \(\bar A \delta \operatorname{Fr}_X U\) it follows that \(\bar A \delta (X \setminus U)\).

Corollary. The Čech extension \(\beta X\) of a completely regular space \(X\) is a perfect bicompact extension.

The following theorem shows that one can give several mutually equivalent definitions of perfect bicompact extensions. As usual, we shall say that a closed set \(F\) splits the space \(X\) into the sets \(U_1\) and \(U_2\) if \(X \setminus F = U_1 \cup U_2\), where \(U_1\) and \(U_2\) are nonempty open sets in \(X\), and \(U_1 \cap U_2 = \Lambda\). We shall say that a set \(N\) (not necessarily closed) splits the space \(Y\) at a point \(y \in N\) if this point has a neighborhood \(U\) such that \(U \cap (Y \setminus N) = V' \cup V''\), where \(V'\) and \(V''\) are open in \(Y \setminus N\), with \(V' \cap V'' = \Lambda\) and \(y \in Y[V'] \cap Y[V'']\).

Theorem 1. Let \(Y\) be a bicompact extension of a completely regular space \(X\). The following properties of the extension \(Y\) are equivalent:

1) the extension \(Y\) is perfect;

2) the remainder \(Y \setminus X\) does not split the bicompactum \(Y\) at any of its points*;

3) for any two disjoint open sets \(U_1\) and \(U_2\) the relation** holds

\[ O\langle U_1 \cup U_2\rangle = O\langle U_1\rangle \cup O\langle U_2\rangle; \]

4) if a closed set \(F\) splits the space \(X\) into the sets \(U\) and \(V\), then the set \(Y[F]\) splits the bicompactum \(Y\) into the sets \(O\langle U\rangle\) and \(O\langle V\rangle\)***.

* \(O_Y\langle U\rangle\) denotes the largest open set of the space \(Y\) that cuts out the set \(U\) on the subspace \(X\), i.e. \(O\langle U\rangle = Y \setminus Y[X \setminus U]\). In what follows, the terminology and results from article (6) will be widely used.

** This result belongs to Yu. M. Smirnov (5).

*** In application to the Čech extension, this property may be regarded as a generalization of the following result of Henriksen and Isbell (7): an open set \(U\) in the Čech extension \(\beta X\) of a space \(X\) is connected if and only if the set \(U \cap X\) is connected. This fact, thus, is valid for every perfect extension.

**** We note that this relation holds for any two open sets if and only if the space is normal and the extension is the Čech extension.

***** Here one cannot discard the requirement that \(Y[F]\) split \(Y\) precisely into the sets \(O\langle U\rangle\) and \(O\langle V\rangle\), since otherwise there would exist imperfect bicompact extensions satisfying this condition.

Theorem 2. In order that a bicompact extension \(Y\) of a space \(X\) be perfect, it is necessary and sufficient that the natural mapping of the Čech extension \(\beta X\) onto the extension \(Y\) be monotone.

The assertion of Theorem 2 is contained in the following two lemmas.

Lemma 2. Let \(Y\) be a perfect bicompact extension of a space \(X\), and let \(Z\) be an arbitrary bicompact extension of the same space following the extension \(Y\); then the natural mapping \(\varphi: Z \to Y\) is monotone.

Lemma 3. Let \(Z\) be a perfect bicompact extension of a space \(X\), and let \(Y\) be a bicompact extension of the same space preceding the extension \(Z\), and suppose that the natural mapping \(\varphi: Z \to Y\) is monotone; then the extension \(Y\) is perfect.

A space \(N\) is called punctiform if every connected bicompact subset of it reduces to a single point. From Theorem 2 and Lemmas 2 and 3 the following theorem follows.

Theorem 3. A space \(X\) has a minimal perfect extension if and only if it has at least one bicompact extension with punctiform remainder. In this case the minimal perfect extension \(\mu X\) is unique, has a punctiform remainder, and is maximal among all bicompact extensions with punctiform remainder.

An obvious consequence of Theorems 1 and 3 is the following strengthening of Duda’s theorem \((^3)\) on the extension of homeomorphisms to bicompact extensions:

Let \(Y_1\) and \(Y_2\) be bicompact extensions of spaces \(X_1\) and \(X_2\), respectively, such that the remainders \(Y_1 \setminus X_1\) and \(Y_2 \setminus X_2\) are punctiform and do not cut any of the bicompacts \(Y_1\) and \(Y_2\) at any point. Then every homeomorphism (if such exists) between the spaces \(X_1\) and \(X_2\) extends to a homeomorphism between the bicompacts \(Y_1\) and \(Y_2\).*

Theorem 3 shows that a space has a unique perfect extension if and only if the remainder in its Čech extension is punctiform. For metrizable spaces this leads to the following result:

Theorem 4. The Čech extension of a metrizable space \(X\) is the unique perfect bicompact extension if and only if \(X=\Phi \cup U\), where \(\Phi\) is compact and \(\dim U=0\).

In proving this theorem the following lemma is used.

Lemma 4. Let \(X\) be a normal space, and let \(X_n\) be the set of points \(x \in X\) such that \(\operatorname{loc}\dim_x X \geq n\).** If the set \(X_n\) is not compact, then the remainder in the Čech extension of the space \(X\) contains a bicompact of dimension \(\geq n\) (in the sense of \(\dim\)).

Now let \(X\) be a peripherally bicompact Hausdorff space. A \(\pi\)-bicompact base on such a space is a base of open sets such that: 1) the boundaries of the sets of this base are bicompact; 2) with every set \(U\) it contains \(X \setminus X[U]\), and with any two sets \(U_1\) and \(U_2\) it contains the sets \(U_1 \cap U_2\) and \(U_1 \cup U_2\). To every \(\pi\)-bicompact base there naturally corresponds a proximity relation on the space \(X\), and the bicompact extension corresponding to this proximity relation has a zero-dimensional (in the sense of \(\operatorname{ind}\)) remainder \((^4)\). Such bicompact extensions (i.e. those corresponding to \(\pi\)-bicompact bases) will for brevity be called \(\pi\)-extensions. Different \(\pi\)-bicompact bases may correspond to one and the same \(\pi\)-extension. To avoid this, one must consider \(\pi\)-bicompact bases satisfying the following additional condi-

* The difference between this formulation and Duda’s theorem consists, first, in the fact that there, if our notation is used, \(Y_1\) and \(Y_2\) are compacta, and, second, in the fact that the condition of “non-cutting” of the bicompacts by the remainders is understood here in a less restrictive sense.

** For the definition of local dimension see \((^2)\).

definition: let \(U\) be an open set of the space \(X\) with bicompact boundary such that for every closed set \(A \subset U\) there exists a set \(V\) from the base containing \(A\) and contained in \(U\); then the set \(U\) itself must belong to the base. The correspondence between such \(\pi\)-bicompact bases and \(\pi\)-extensions is one-to-one.

Among the \(\pi\)-extensions of the space \(X\), there is, obviously, a unique maximal one—the extension corresponding to the \(\pi\)-bicompact base consisting of all open sets with bicompact boundary.

Theorem 5. The maximal \(\pi\)-extension is perfect.

It follows directly from Theorems 3 and 5 that the maximal \(\pi\)-extension of a peripherally bicompact space \(X\) coincides with the minimal perfect extension \(\mu X\) and, consequently, is maximal among extensions with pointiform (in particular, with zero-dimensional) remainder.

The following two theorems give, in a certain sense, a description of the bicompact extensions of a peripherally bicompact space that follow the maximal \(\pi\)-extension and that precede it.

Theorem 6. A bicompact extension \(Y\) of a peripherally bicompact space \(X\) follows the maximal \(\pi\)-extension \(\mu X\) if and only if it is perfect relative to all open sets of the space \(X\) that have bicompact boundary.

Theorem 7. A bicompact extension \(Y\) of a peripherally bicompact space \(X\) precedes the maximal \(\pi\)-extension \(\mu X\) if and only if it corresponds to some uniform structure of finite open covers consisting of sets with bicompact boundary.

In conclusion we apply the preceding results to obtain an answer to the following question from the survey of P. S. Aleksandrov (1): does every peripherally bicompact space possess a bicompact extension with zero-dimensional remainder having the same dimension as the space itself? The answer turns out to be negative. Let \(X\) be the set of points of the unit square in the plane for which at least one coordinate is rational, and let \(Y\) be the closure of the space in the plane (i.e., the unit square). From Theorems 1 and 3 it follows that all bicompact extensions of the space \(X\) with zero-dimensional remainder precede the extension \(Y\), whence, in turn, it follows that all of them have dimension \(\geqslant 2\), whereas the dimension of the space \(X\) is equal to 1.

Moscow State University
named after M. V. Lomonosov

Received
6 X 1960

References

1. P. S. Aleksandrov, UMN, 15, 2, 25 (1960).
2. C. H. Dowker, Quart. J. Math., 6, 101 (1955).
3. R. Duda, Indagationes Math., 22, 2, 132 (1960).
4. E. Sklyarenko, DAN, 120, No. 6, 1200 (1958).
5. Yu. M. Smirnov, “Matem. sborn.”, 29, 1, 157 (1951).
6. Yu. M. Smirnov, Matem. sborn., 31, 3, 543 (1952).
7. M. Henriksen, J. R. Isbell, Illinois J. Math., 1, 574 (1957).