UDC 513.83

MATHEMATICS

E. G. SKLYARENKO

UNIFORM STRUCTURES OF BORDERINGS

(Presented by Academician L. I. Sedov on 11 X 1968)

A finite system of open sets \(U_i\) is called a bordering of a completely regular space \(X\) if the set \(X \setminus \bigcup_i U_i\) is bicompact. The concept of a bordering was introduced by Yu. M. Smirnov \((^{2,3})\) for the characterization of spaces possessing bicompact extensions whose remainders have a prescribed finite dimension. Yu. M. Smirnov showed that (under certain restrictions) a completely regular space \(X\) possesses a bicompact extension \(Y\), for which \(\dim (Y \setminus X) \leq n\), if and only if there is on it a structure of borderings of multiplicity \(\leq n+1\) possessing the base property (see below).

We shall call a system of borderings \(\Omega\) of a completely regular space \(X\) a uniform structure of borderings if \(\Omega\) satisfies the following conditions:

1. If \(\xi \in \Omega\) and \(\eta < \xi\) (i.e., every set of \(\xi\) is contained in some set of \(\eta\)), then \(\eta \in \Omega\).

2. If \(\xi \in \Omega\) and \(B\) is a bicompact subset of \(X\), then the restriction of \(\xi\) to the subspace \(X \setminus B\) is contained in \(\Omega\).

3. For any \(\xi, \eta \in \Omega\) there is a \(\zeta \in \Omega\) such that \(\xi \wedge \eta < *\zeta\) (\(\xi \wedge \eta\) is the system of sets of the form \(U \cap V\), \(U \in \xi\), \(V \in \eta\); the symbol \(\xi < *\eta\) means that for every point \(x \in X\) there exists a \(U \in \xi\) such that the star \(O_\eta x\) of the point \(x\) with respect to the system of sets \(\eta\) is contained in \(U\)).

4. For every point \(x \in X\) and every neighborhood \(Ox\) of it there exists a bordering \(\xi \in \Omega\) such that every \(U \in \xi\) whose closure \([U]\) contains \(x\) is contained in \(Ox\).

A system of borderings satisfying conditions 3, 4 will be called the confinal part of a (corresponding) uniform structure of borderings. It is easy to verify that conditions 3, 4 determine the structure uniquely. We note that in the case of open coverings, conditions 1, 3, and 4 may be taken as the basis of the definition of the uniform structure of a topological space (for the definition of the uniform structure of open coverings, see, for example, \((^{1})\)). It is not hard to verify that condition 4 is equivalent to the following requirement.

\(4'.\) For every point \(x \in X\) and every neighborhood \(Ox\) of it there exists a neighborhood \(O_1x\) and a bordering \(\xi \in \Omega\) such that \(O_\xi O_1x \subset Ox\).

A system of borderings satisfying conditions 3 and \(4'\) is called by Yu. M. Smirnov a structure of borderings possessing the base property; thus, Yu. M. Smirnov’s structure of borderings coincides with the confinal part of a uniform structure of borderings in the sense indicated above.

The set of all uniform structures of borderings of the space \(X\) is partially ordered: as usual, we regard \(\Omega < \Omega_1\) if the system \(\Omega\) is contained in \(\Omega_1\). The aim of the present note is to describe the partially ordered set of all uniform structures of borderings of an arbitrary completely regular space \(X\).

Let \(\Omega_0\) be the subsystem in \(\Omega\) consisting of all coverings \(\alpha \in \Omega\).

Lemma 1. For every uniform structure of borderings \(\Omega\), the system of coverings \(\Omega_0\) is a uniform structure of the space \(X\).

From this lemma there obviously follows

Corollary. The correspondence \(\omega:\Omega\to\Omega_0\) defines an order-compatible mapping of the partially ordered set of all uniform structures of fringes of the completely regular space \(X\) onto the set of all uniform structures of finite open coverings of the space \(X\).

We shall say that the uniform structure of fringes \(\Omega\) is compatible with the uniform structure of coverings \(\Omega_0\) if \(\Omega_0=\omega(\Omega)\). Let \(Y\) be the bicompact extension of the space \(X\) corresponding to the uniform structure \(\Omega_0\) (see (1)). For every open set \(U\) of the space \(X\), denote by \(O(U)\) the set \(Y\setminus [X\setminus U]\). Obviously, \(O(U)\cap X=U\); \(O(U)\) is maximal among all open sets in \(Y\) that cut \(U\) on \(X\). Denote by \(O_N(U)\) the intersection of \(O(U)\) with the remainder \(N=Y\setminus X\). From Lemma 1 it follows easily (see below) that for every fringe \(\xi\) from the uniform structure \(\Omega\), compatible with \(\Omega_0\), the system of sets \(O_N(U)\), \(U\in\xi\), is a covering of the remainder \(N\). Denote by \(\Sigma\) the system of all coverings of \(N\) obtained in this way from fringes of the structure \(\Omega\), and by \(\Sigma_0\) the uniform structure of the space \(N\) obtained in the same way from \(\Omega_0\).

Theorem. The correspondence \(\sigma:\Omega\to\Sigma\) carries out an isomorphism of the set of all uniform structures of fringes of the completely regular space \(X\), compatible with the structure of finite open coverings \(\Omega_0\), onto the set of all uniform structures of finite open coverings of the remainder \(N\) that follow the uniform structure \(\Sigma_0\). If \(N\) is dense in \(Y\) (i.e., if \(X\) has no points of local bicompactness), then \(\sigma\) is a mapping onto the whole set of uniform structures of coverings of \(N\) that follow \(\Sigma_0\). Moreover, in this case the operator \(O_N(\ )\) carries out an isomorphism of the set of all uniform structures of coverings of the space \(X\) that follow \(\Omega_0\), onto the set of all uniform structures of fringes of \(N\) compatible with the structure \(\Sigma_0\). The operator \(O_X(V)\), \(V\subset N\), gives the inverse isomorphisms.

Thus the concept of a uniform structure of fringes is not only a generalization of the concept of a uniform structure of coverings, but also, in a certain sense, a dual concept.

We now pass to the proof of the formulated assertions. The following proposition is a strengthening of condition 4.

Lemma 2. For any bicompact set \(B\subset X\) and for any neighborhood \(W\) of the set \(B\), there is a fringe \(\xi\in\Omega\) such that every \(U\in\xi\) whose closure meets \(B\) is contained in \(W\).

Proof. For every point \(x\in B\) choose a fringe \(\xi(x)\in\Omega\) such that \(U\subset W\), if \(x\in[U]\), \(U\in\xi(x)\). Obviously, the point \(x\) has a neighborhood \(B(x)\) in \(B\) such that from \([U]\cap B(x)\ne\varnothing\), \(U\in\xi(x)\), it follows that \(U\subset W\). From the covering \(\{B(x)\}\) of the bicompact \(B\) choose a finite subcovering \(B(x_1),\ldots,B(x_n)\). By conditions 1, 3 the fringe \(\xi=\xi(x_1)\wedge\cdots\wedge\xi(x_n)\) belongs to \(\Omega\). The fringe \(\xi\) satisfies the requirement of the lemma.

Proof of Lemma 1. For every fringe \(\xi\in\Omega\) denote by \(B_\xi\) the bicompact \(X\setminus\bigcup_i U_i\), \(U_i\in\xi\). By condition 1, the covering \(\alpha\) that includes \(\xi\) and any finite collection of open sets covering \(B_\xi\) belongs to \(\Omega_0\); therefore the system \(\Omega_0\) is nonempty. The system \(\Omega_0\) obviously satisfies condition 1. Condition 4 is also fulfilled for \(\Omega_0\): first choose a fringe \(\xi\in\Omega\) satisfying this condition, and then add to it the set \(Ox\) and any open set \(W\) containing \(B_\xi\setminus Ox\) but whose closure does not contain the point \(x\); the resulting covering is contained in \(\Omega_0\) and satisfies the needed requirements. By condition 3 the covering \(\alpha\wedge\beta\) belongs to \(\Omega_0\), if \(\alpha,\beta\in\Omega_0\). Therefore it remains only to prove that for any \(\alpha\in\Omega_0\) there exists \(\beta\in\Omega_0\) such that \(\alpha<_* \beta\).

Let \(\alpha \in \Omega_0\) be an arbitrary cover. In accordance with condition 3, there is a \(\xi \in \Omega\) such that \(\alpha < *\xi\). Let \(\alpha'\) be a cover of the Čech extension \(\beta X\), consisting of the sets \(O(U)\), \(U \in \alpha\), and of the set \(\beta X \setminus B_\xi\); let \(\gamma\) be a finite open cover of \(\beta X\), star-refined into \(\alpha'\), and let \(\bar\gamma\) be a closed cover of \(\beta X\) refined into \(\gamma\). Let \(\bar B_1,\ldots,\bar B_k\) be those sets of \(\bar\gamma\) which have nonempty intersection with \(B_\xi\), and let \(\Gamma_1,\ldots,\Gamma_k\) be elements of \(\gamma\) containing them; denote \(\bar B_i \cap B_\xi\), \(\Gamma_i \cap X\), respectively, by \(B_i, V_i\). The sets \(B_i\) cover \(B_\xi\), and the system of sets \(\nu=\{V_i\}\) is star-refined into \(\alpha\).

By Lemma 2, for each \(B_j\) there is a border \(\xi_j \in \Omega\) such that from \([U]\cap B_j \ne \varnothing\), \(U \in \xi_j\), it follows that \(U \subset V_j\). Let \(\eta \in \Omega\) be the border equal to the intersection of all the borders \(\xi,\xi_1,\ldots,\xi_k\). From the fact that \(U \in \eta\) and \([U]\cap B_\xi \ne \varnothing\), it follows, obviously, that \(U \subset V_j\) for some \(j\). Let \(C\) be the closure of the union of all those \(U \in \eta\) whose closures do not intersect \(B_\xi\); let \(\nu'\) be the system of sets \(V'_j=V_j\setminus C\), and let \(W=\bigcup_{j=1}^k V'_j\). By construction we have \(\alpha < *(\nu'\cup \eta)\); however, the system of sets \(\nu'\cup\eta\) is not a cover (the points of the set \(B_\eta\setminus W\) are not covered).

Let \(W_1\) be a neighborhood of the set \(B_\xi\) such that \([W_1]\subset W\). Denote by \(\xi'\) the restriction of the system \(\xi\) to \(X\setminus [W_1]\), by \(\nu''\) the restriction of \(\nu'\) to \(W_1\), and by \(\xi''\) the intersection with \(\nu'\) of the restriction of the system \(\xi\) to the set \(W\setminus B_\xi\). Put \(\beta=\nu''\cup\eta\cup\xi'\cup\xi''\). Obviously, \(\beta\) is a finite open cover of \(X\). Since \(\beta<\eta\) and \(\eta\in\Omega\), we have \(\beta\in\Omega_0\). For every point \(x\in W_1\), the star \(O_{\beta x}\) consists of sets contained in sets of the system \(\nu\), and for every point \(x\in X\setminus W_1\), of sets contained in sets of the system \(\xi\). Therefore \(\alpha<*\beta\). The lemma is proved.

It follows from this lemma that, for any \(\xi\in\Omega\), the system of sets
\[ O_N(\xi)=\{O_N(U),\, U\in\xi\} \]
covers \(N=Y\setminus X\). Indeed, let \(y\) be an arbitrary point of \(N\), and let \(O_y\) be a neighborhood in \(Y\) whose closure does not intersect \(B_\xi\). Then the cover of \(X\) consisting of the elements of \(\xi\) and of the set \(X\setminus[O_y\cap X]\) belongs to \(\Omega_0\) and, consequently, extends to a cover of \(Y\), and moreover \(y\), obviously, does not belong to \(O(X\setminus[O_y\cap X])\).

Proof of the theorem. We begin with some auxiliary observations. Recall that an open set \(U\) is called canonical if
\[ U=\operatorname{Int}[U]=X\setminus [X\setminus[U]]. \]
We shall call a border \(\xi\in\Omega\) canonical if every \(U\in\xi\) is a canonical open set of the space \(X\setminus B_\xi\). For every \(\xi\in\Omega\), denote by \(\bar\xi\) the border of \(X\) obtained by replacing each \(U\in\xi\) by the minimal canonical open (in \(X\setminus B_\xi\)) set \(\bar U\) containing \(U\). Since \(\bar\xi<\xi\), we have \(\bar\xi\in\Omega\). If \(\xi<*\eta<*\zeta\), then \(\bar\xi<\bar\zeta\) (since the cover \(\zeta\) of the space \(X\setminus B_\zeta\) is refined by \(\xi\)). Therefore the subsystem \(\overline{\Omega}\subset\Omega\), consisting of all canonical borders, is a cofinal part of the structure \(\Omega\). Obviously, if \(\xi\in\overline{\Omega}\), then for any bicompact set \(B\subset X\) the restriction of \(\xi\) to \(X\setminus B\) also belongs to \(\overline{\Omega}\).

For every \(U\in\xi\), \(\xi\in\Omega\), the set \(O(U)\), being the maximal open set cutting out \(U\) on \(X\), is a canonical open set in \(Y\setminus B_\xi\). If \(\eta\) is the restriction of \(\xi\) to \(X\setminus B\), where \(B\) is a bicompact set, then \(O_N(\eta)=O_N(\xi)\). Further, the relations \(\xi<\eta\), \(\xi<*\eta\) imply, obviously, the relations \(O_N(\xi)<O_N(\eta)\), \(O_N(\xi)<*O_N(\eta)\) (\(\xi,\eta\) arbitrary). The system of covers
\[ O_N(\Omega)=\{O_N(\xi),\, \xi\in\Omega\} \]
satisfies condition 4, since this condition is satisfied by the system of covers \(O_N(\overline{\Omega}_0)\) contained in it. For every uniform structure of covers \(\Omega'_0\), for which \(\Omega_0<\Omega'_0\), \(O_N(\overline{\Omega}'_0)\) is a system of borders of the remainder \(N\), also, obviously, satisfying condition 4.

Let \(N\) be dense in \(Y\). Then for every open \(U\subset X\) we have the inclusion
\[ U\subset O_X(O_N(U)). \]
Moreover, if \(U\in\xi,\ \xi\in\overline{\Omega}\), then \(U=\)

\(= O_X(O_N(U)) \cap (X \setminus B_\xi)\). If, however, \(\xi\) is a cover of \(X\) consisting of a finite number of canonical sets, then \(\xi = O_X(O_N(\xi))\). Taking into account that it suffices to consider only canonical covers and bordifications, we obtain the assertion of the theorem for the case when \(N\) is dense in \(Y\).

To prove Theorem 1 it remains to consider the special case when \(Y\) is not equal to the closure of \(N\). We have seen that the system of covers \(O_N(\Omega)\) of the set \(N\) satisfies conditions 3 and 4. Therefore it is the cofinal part of some uniform structure of finite covers of \(N\) (obviously following \(\Sigma_0\)), which we shall put in correspondence with the structure of bordifications \(\Omega\). The fact that different bordification structures of \(X\) will correspond to different cover structures of \(N\) follows from the following assertion: from \(O_N(\gamma) < O_N(\xi)\) and \(\xi \in \Omega\) it follows that \(\gamma \in \Omega\) (\(\gamma\) is any bordification). For the proof take bordifications \(\eta,\zeta \in \Omega\) such that \(\xi < *\eta < *\zeta\). Let \(Y' = Y \setminus (B_\gamma \cup B_\xi)\), and let \(\gamma', \xi', \zeta'\) be covers of \(Y'\) obtained by extending to \(Y'\) the bordifications \(\gamma, \xi, \zeta\). For any \(U' \in \xi'\) choose \(V' \in \xi'\) such that \([U'] \subset V'\) (closure in \(Y'\)), and \(W' \in \gamma'\) such that \(V' \cap N \subset W' \cap N\). The set \([U'] \setminus W'\), closed in \(Y'\), does not intersect \(N\) (otherwise the set \(V' \cap N\) would not be contained in \(W' \cap N\)). Summing the sets of the form \([U'] \setminus W'\), constructed in this way for all \(U' \in \xi'\), we obtain a set \(C\), closed in \(Y'\), which does not intersect \(N\). Let \(B = B_\gamma \cup B_\xi \cup C\); the set \(B\) is bicompact and lies in \(X\). The restriction of the bordification \(\zeta\) to \(X \setminus B\) is inscribed in \(\gamma\). Therefore, by conditions 1, 2, \(\gamma \in \Omega\).

Remark 1. If the extension \(Y\) has a countable base, then \(O_N(\ )\) maps the uniform bordification structures compatible with \(\Omega_0\) onto all uniform structures of finite covers of \(N\) following \(\Sigma_0\). It can be shown that in the general case this is not so.

Remark 2. The assertion of Lemma 1 is valid for a system \(\Omega\) of arbitrary (not necessarily finite) bordifications satisfying conditions 1, 2, 3, and \(4'\); in the proof, however, one must make the obvious changes caused by replacing requirement 4 by condition \(4'\) of Yu. M. Smirnov (for example, in Lemma 2 there will be found such a \(\xi \in \Omega\) and such a neighborhood \(V \supset B\) that every \(U \in \xi\) for which \(U \cap V \ne \varnothing\) is contained in \(W\)).

REFERENCES

1. Yu. M. Smirnov, Mat. sborn., 31, 283 (1956).
2. Yu. M. Smirnov, Mat. sborn., 69, 141 (1966).
3. Yu. M. Smirnov, Mat. sborn., 71, 454 (1966).