Mathematics

L. A. Skornyakov

On Compact Topological Spaces

(Presented by Academician A. I. Mal'tsev on 26 V 1961)

It is well known that compact spaces can be represented (the terminology of Bourbaki, \((^3)\), pp. 108–109, is used) by inverse spectra of finite discrete spaces (see, for example, \((^1)\), pp. 50 and 51). In the present note a construction is described which makes it possible to obtain arbitrary compact spaces as direct spectra of finite discrete spaces.

Let \(\Omega\) be a directed set. To each \(\alpha \in \Omega\) there is assigned a set \(M_\alpha\) (the intersection \(M_\alpha \cap M_\beta\), for \(\alpha \ne \beta\), is regarded as empty). On the set \(M = \bigcup M_\alpha\) a reflexive and symmetric relation \(\Delta\) is given. If \(x \in M_\alpha\), then \(\alpha\) will be called the index of the element \(x\) and will be denoted by \(Ix\). We shall say that \(x \equiv y\;(\alpha)\) \((x, y \in M,\ \alpha \in \Omega)\) if there exists an element \(z \in M\) such that \(Iz \geq \alpha,\ x \Delta z\), and \(y \Delta z\).

The aggregate \(\mathfrak S = (\Omega, M, \Delta)\) will be called a spectrum if property C holds:

Property C. For every \(\alpha \in \Omega\) there exists an index \(\beta \in \Omega\) such that from the relations \(x \equiv y\;(\beta)\), \(y \equiv z\;(\beta)\), and \(Iy > \beta\) it follows that \(x \equiv z\;(\alpha)\) (this index \(\beta\) will be denoted by \(\alpha/2\)).

Remark 1. For every \(\alpha \in \Omega\) there exists an index \(\beta > \alpha\) (we shall denote it by \(\alpha/4\)) such that from the relations \(x \Delta y\), \(y \equiv z\;(\beta)\), \(z \Delta u\), \(Iy, Iz > \beta\), it follows that \(x \equiv u\;(\alpha)\).

Indeed, since \(x \equiv y\;(\beta)\) and \(z \equiv u\;(\beta)\), one may take as \(\alpha/4\) any index exceeding \(\alpha\), \(\alpha/2\), and \((\alpha/2)/2\).

We shall call a thread such a nonempty subset \(S\) of the set \(M\) that:

B1. The intersection \(S \cap M_\alpha\) is either empty or contains exactly one element \(s_\alpha\).

B2. If \(\beta > \alpha\) and \(s_\alpha\) exists, then \(s_\beta\) also exists.

B3. If \(s_\alpha\) and \(s_\beta\) exist, then \(s_\alpha \Delta s_\beta\).

We shall say that the threads \(S\) and \(T\) are close of order \(\alpha\) (in notation \(S \equiv T\;(\alpha)\)) if \(s_\beta \equiv t_\gamma\;(\alpha)\) for all \(\beta\) and \(\gamma\) for which \(s_\beta\) and \(t_\gamma\) exist.

From this and from Remark 1 it follows:

Remark 2. If \(s_\beta \equiv t_\gamma\;(\alpha/4)\) for some \(\beta, \gamma > \alpha/4\), then \(S \equiv T\;(\alpha)\).

Theorem 1. The relation \(S \equiv T\;(\alpha)\) defines a uniform structure on the set of threads.

For the proof it is enough to verify the validity of conditions \(1^\circ\)—\(4^\circ\), indicated in \((^3)\) on p. 151. The first three of them are obvious. To prove the last, choose \(\beta > (\alpha/4)/2\) and suppose that \(R \equiv S\;(\beta)\) and \(S \equiv T\;(\beta)\). Then \(r_\beta \equiv t_\beta\;(\alpha/4)\) and, in view of Remark 2, \(R \equiv T\;(\alpha)\).

In view of \((^3)\), Ch. II, § 1 and 4, with the constructed uniform structure there is associated a separated uniform structure, and hence also a separated uniform space (\((^3)\), p. 162, Proposition 1). This uniform pro-

space will be called the limit of the spectrum \(\mathfrak S\) and denoted by \(\mathfrak A(\mathfrak S)\), or by \(\mathfrak A(\Omega,M,\Delta)\). The elements of the space \(\mathfrak A(\mathfrak S)\) will be called pencils.

Theorem 2. If all \(M_\alpha\) are finite, then for the precompactness of the space \(\mathfrak A(\mathfrak S)\) it is sufficient that the following hold:

Condition II. For every \(\alpha\in\Omega\) and every pencil \(\mathfrak f\) there is a fan \(S\in\mathfrak f\) such that \(s_\alpha\) exists.

Indeed, let \(\alpha\in\Omega\) and \(M_\alpha=\{x^1,\ldots,x^n\}\). Denote by \(U^i\) the set of those pencils each of which contains a fan \(S\) for which \(s_\alpha=x^i\). If \(\mathfrak f,\mathfrak t\in U^i\), then for suitable fans \(S\in\mathfrak f\) and \(T\in\mathfrak t\) we shall have \(s_\alpha=t_\alpha=x^i\). Applying B3, we see that \(S\equiv T\ (\alpha)\), i.e., that \(\mathfrak f=\mathfrak t\ (\alpha)\). Consequently, the set \(U^i\) is small of order \(\alpha\). In view of condition II, the system \(U^1,\ldots,U^n\) forms a covering of the space. It remains to apply Theorem 4 from \((^3)\) (p. 182).

Let now \(E\) be a uniform space and \(\alpha\) a certain entourage \((^3)\), p. 159). A subset \(G\) of the space \(E\) will be called an \(\alpha\)-net if for every \(a\in E\) there is an element \(b\in G\) such that \([a,b]\alpha\). If \(\Omega\) is a filter of entourages of the space \(E\), then every set of the form \(G=\bigcup G_\alpha\), where \(G_\alpha\) is some fixed \(\alpha\)-net, will be called a net space of the space \(E\). It is clear that \(\overline G=E\).*

We next consider a spectrum \(\mathfrak S=(\Omega,M,\Delta)\). We shall call this spectrum proper if for every \(x\in M\) there is a fan \(S\) containing \(x\). Thus, in a proper spectrum, for every \(x\in M\) one may choose a fan \(S\) containing \(x\), and denote by \(\varphi(x)\) the pencil containing the fan \(S\). The mapping \(\varphi\) thus obtained from the set \(M\) into the space \(\mathfrak A(\mathfrak S)\) will be called injective. It is easy to verify that \(\varphi(M)\) is a net space.

Taking Remark 2 into account, it is not difficult to verify the validity of the following properties:

И1. If \(x\equiv y\ (\alpha/4)\), \(l x,\ l y>\alpha/4\), then \([\varphi(x),\varphi(y)]\alpha\).

И2. If \([\varphi(x),\varphi(y)]\alpha\), then \(x\equiv y\ (\alpha)\).

Theorem 3. Every separable compact space is isomorphic to some \(\mathfrak A(\Omega,M,\Delta)\), where all \(M_\alpha\) are finite, while the spectrum itself is proper and satisfies condition II.

Proof. Let \(E\) be an arbitrary compact space. It may be regarded as uniform \((^3)\), p. 178, Theorem 2). Denote by \(\Omega\) the collection of symmetric open entourages of the corresponding uniform structure. If \(\alpha,\beta\in\Omega\), put \(\alpha\geq\beta\) if from \([a,b]\alpha\) it follows that \([a,b]\beta\). It is easy to see that \(\Omega\) becomes a directed set. Let \(M_\alpha\) be an \(\alpha\)-net of the space \(E\). We may assume that all \(M_\alpha\) are finite \((^3)\), p. 181, Theorem 3). Let \(M=\bigcup M_\alpha\) (points coinciding in \(E\), but belonging to different \(M_\alpha\), are regarded as distinct). If \(x\in M_\alpha,\ y\in M_\beta\), then put \(x\Delta y\) in the case when the intersection \(\alpha(x)\cap\beta(y)\)*** is nonempty.

(a) For every \(\alpha\in\Omega\) there exists \(\beta\in\Omega\) such that for any two points \(a,b\in E\) close of order \(\beta\), there exists \(x\in M_\alpha\) such that \(a,b\in\alpha(x)\).

Indeed, suppose that for every \(\beta\in\Omega\) there are points \(a_\beta\) and \(b_\beta\), close of order \(\beta\), such that for every \(x\in M_\alpha\) either \(a_\beta\notin\alpha(x)\) or \(b_\beta\notin\alpha(x)\). Let \(A_\beta=\{a_\gamma;\ \gamma>\beta\}\). It is easy to understand that the system \(\{A\}\) is centered. Let \(c\in\bigcap A_\beta\). But \(c\in\alpha(x)\) for some \(x\in M_\alpha\). Find \(\lambda\) such that \(\lambda(c)\subset\alpha(x)\). Let \(\mu>\lambda\)****. For some \(\nu>\mu\)

* The notation \([a,b]\alpha\) means that \(a\) and \(b\) are close of order \(\alpha\).

** By \(\overline X\) is denoted the closure of the set \(X\).

*** \(\lambda(c)=\{x;\ x\in E;\ x\equiv c(\lambda)\}\) (cf. \((^3)\), p. 158).

**** We shall say that \([a,b]\alpha\circ\beta\) if there exists a point \(c\) such that \([a,c]\alpha\) and \([c,b]\beta\); put \({}^2\alpha=\alpha\circ\alpha\) (cf. \((^3)\), p. 152).

find \(a_\nu \in \mu(c)\). Then \(b_\nu\) and \(c\) turn out to be close of order \(\lambda\), i.e. \(b_\nu \in \lambda(c) \subset \alpha(x)\). But \(a_\nu \in \mu(c) \subset \lambda(c) \subset \alpha(x)\). A contradiction.

It is not hard to verify that as \(\alpha/2\) one may take such a \(\gamma\) that \(\gamma \overset{6}{>} \beta\), where \(\beta\) is the index found in assertion (a). Thus, \(\mathfrak S(\Omega,M,\Delta)\) turns out to be a spectrum. It is easy to verify that this spectrum is proper.

Let \(a \in E\). For each \(\alpha \in \Omega\) choose \(s_\alpha \in M_\alpha\) so that \(a \in \alpha(s_\alpha)\). Then \(S=\{s_\alpha\}\) turns out to be a fan. We shall denote the corresponding bunch by \(\mathfrak f(a)\). If \(S\) is an arbitrary fan, put \(S^\alpha=\{s_\beta;\ \beta>\alpha\}\). The system \(\{S^\alpha\}\) turns out to be a base of a Cauchy filter in \(E\). In view of Theorem 1 from \((^3)\) (p. 178), this filter has a limit \(e(S)\). It is easy to see that \(S\in \mathfrak f(e(S))\). This proves that the spectrum \(\mathfrak S\) satisfies condition II. Hence it is also clear that the mapping \(a\to \mathfrak f(a)\) is a mapping of \(E\) onto \(\mathfrak A(\mathfrak S)\). It is easy to verify that it is one-to-one. From assertion (a) it follows that it is uniformly continuous. Application of Theorem 2 and Corollary 2 from \((^3)\) (pp. 181 and 114) shows that the spaces \(E\) and \(\mathfrak A(\mathfrak S)\) are isomorphic.

Construction. Let a spectrum \(\mathfrak S=(\Omega,M,\Delta)\) satisfy condition II, and let the spectrum \(\mathfrak S'=(\Omega,M',\Delta')\) be proper. Suppose that a not necessarily single-valued mapping \(f\) of the set \(M\) into the set \(M'\) is given, with:

1. \(Ix=Ix'\) for all \(x'\in f(x)\).

2. For every \(\alpha\in\Omega\) there exists an index \(\beta\in\Omega\) such that \(Ix,Iy>\beta\) and \(x\equiv y(\beta)\) imply \(x'\equiv y'(\alpha)\) for any \(x'\in f(x)\) and \(y'\in f(y)\).

Let, further, \(\mathfrak f\) be a bunch from \(\mathfrak A(\mathfrak S)\) and \(K'=\bigcup_{S\in \mathfrak f} f(S)\). If \(x'\in K'\), then, in view of the properness of the spectrum \(\mathfrak S'\), there exists a fan \(X'\) containing \(x'\). We shall denote the corresponding bunch by \(\mathfrak x'\). Put \(\mathfrak F^\alpha=\{\mathfrak x';\ Ix'>\alpha\}\). Since \(\mathfrak S\) satisfies condition II, \(\{\mathfrak F^\alpha\}\) turns out to be a base of a filter. From 01 and 02 it is not hard to derive that this filter is a Cauchy filter. In view of \((^3)\) (p. 172, Theorem 2),

\[ \lim \mathfrak F^\alpha = F(\mathfrak f) \]

belongs to the completion \(\overline{\mathfrak A(\mathfrak S')}\) of the space \(\mathfrak A(\mathfrak S')\).

Remark 3. For every \(\alpha\in\Omega\) there exists an index \(\beta\in\Omega\) such that, for all \(x\in M\) for which \(Ix>\beta\), one has \([F(\mathfrak f),\varphi'(x')]\alpha\), where \(x'\) is any element of \(f(x)\); \(\mathfrak f\) is any bunch containing such a fan \(S\) that \(x\in S\); \(\varphi'\) is an arbitrary injection mapping.

Indeed, let \(\gamma \overset{2}{>} \alpha\) and \(\beta>\gamma\) be such that \([F(\mathfrak f),\mathfrak x']\gamma\) for all \(\mathfrak x'\in \mathfrak F^\beta\). If \(Ix>\beta\), then, in view of 01, \(Ix'>\beta\). Therefore \([\mathfrak x',\varphi'(x')]\beta\). Since \(\mathfrak x'\in \mathfrak F^\beta\), it follows that \([F(\mathfrak f),\varphi'(x')]\alpha\).

Theorem 4. The mapping \(F\) constructed in the construction described above is a uniformly continuous mapping of the space \(\mathfrak A(\mathfrak S)\) into \(\overline{\mathfrak A(\mathfrak S')}\). Every uniformly continuous mapping \(\Phi\) of the space \(\mathfrak A(\mathfrak S)\) into \(\overline{\mathfrak A(\mathfrak S')}\) can be obtained in this way by using a single-valued mapping \(f\).

Proof. Let \(\alpha\in\Omega\). Denote by \(\beta\) an index exceeding the indices indicated for \(\alpha/4\) in Remark 3 and in condition 02. If \([\mathfrak f,\mathfrak t]\beta,\ S\in\mathfrak f,\ T\in\mathfrak t\), then for \(\lambda,\mu>\beta\) we shall have \(s_\lambda\equiv t_\mu(\beta)\), and hence \(x'\equiv y'(\alpha/4)\) for any \(x'\in f(s_\lambda)\), \(y'\in f(t_\mu)\). Moreover, by Remark 3, \([F(\mathfrak f),\varphi'(x')]\alpha\) and \([F(\mathfrak t),\varphi'(y')]\alpha\). Taking \(I1\) into account, we obtain \([F(\mathfrak f),F(\mathfrak t)]\alpha\). This proves the uniform continuity of the mapping \(F\).

Now suppose a mapping \(\Phi\) is given. Denote by \(\varphi\) and \(\varphi'\) some injection mappings in the spectra \(\mathfrak S\) and \(\mathfrak S'\), respectively. If \(x\in M\), then \(\Phi(\varphi(x))\in \overline{\mathfrak A(\mathfrak S')}\). Denote by \(f(x)\) some element of \(M'_{Ix}\) satisfying the relation \([\varphi'(f(x)),\Phi(\varphi(x))]\overset{2}{Ix}\). It is clear that condition 01 is fulfilled for the mapping \(f\). Let, further, \(\alpha\in\Omega\) and \(\gamma\overset{5}{>}\alpha\). Choose-

choose \(\beta>\gamma\) so that from \([a,b]\,\beta\) it follows that \([\Phi(a),\Phi(b)]\,\gamma\). If \(Ix,Iy>\beta/4\) and \(x\equiv y\,(\beta/4)\), then, by I1, \([\Phi(\varphi(x)),\Phi(\varphi(y))]\,\gamma\). Therefore \([\varphi'(x'),\varphi'(y')]\,\gamma\), and I2 gives \(x'\equiv y'(\alpha)\). Thus the mapping \(f\) has property 02. Using the construction, we shall build the mapping \(F\). Since \(\varphi(\overline{M})=\mathfrak A(\mathfrak S)\), then, in view of \((^3)\) (p. 171, Theorem 1; p. 72, corollary), in order to prove the coincidence of the mappings \(F\) and \(\Phi\) it is enough to establish that
\[ F(\varphi(x))=\Phi(\varphi(x)) \]
for all \(x\in M\). To verify this, take in the bundle \(\varphi(x)\) some fan \(S\). Let \(\beta\in\Omega\) and let \(\gamma>\beta\) be such that from \([a,b]\,\gamma\) there follow the relations \([F(a),F(b)]\,\beta\) and \([\Phi(a),\Phi(b)]\,\beta\). Suppose further that the index \(\delta\) exceeds \(\beta,\gamma/4\) and the index indicated for \(\gamma\) in Remark 3. From Remarks 2 and 3 it follows that \([\varphi(x),\varphi(s_\delta)]\,\gamma\) and \([F(\varphi)(s_\delta),\varphi'(f(s_\delta))]\,\gamma\). Hence, taking into account that, by the construction of the mapping \(f\), one has
\[ [\varphi'(f(s_\delta)),\Phi(\varphi(s_\delta))]\,\delta, \]
we obtain
\[ [F(\varphi(x)),\Phi(\varphi(x))]\,\beta\circ\gamma\circ\delta\circ\beta. \]
Since \(\beta\circ\gamma\circ\delta\circ\beta>\beta\), and \(\beta\) is arbitrary, everything is proved.

Theorem 5. Let \(\mathfrak S=(\Omega,M,\Delta)\) be a regular spectrum satisfying condition \(\Pi\), and let all \(M_\alpha\) be abstract algebras ((\(^{2}\), p. 7) with one and the same system \(\mathfrak G\) of operations; moreover, for each \(g\in\mathfrak G\) one has
\[ g(x_1,\ldots,x_m)\,\Delta\,g(y_1,\ldots,y_m), \]
if \(x_i\Delta y_i\) for all \(i\). Then the space \(\overline{\mathfrak A(\mathfrak S)}\) is an abstract algebra with the same system of operations.

For the proof it should be noted that every \(g\in\mathfrak G\), as a mapping of
\[ \underbrace{M\times\cdots\times M}_{m\ \text{times}} \]
into \(M\), satisfies conditions 01 and 02. After this it remains to apply Theorem 4.

Moscow State University
named after M. V. Lomonosov

Received
9 V 1961

REFERENCES

\(^{1}\) P. S. Aleksandrov, UMN, 2, No. 1 (17) (1947). \(^{2}\) G. Birkhoff, Theory of structures, Moscow, 1952. \(^{3}\) N. Bourbaki, General Topology (Basic Structures), Moscow, 1958.