UDC 513.83

MATHEMATICS

M. A. GOLDMAN

A CHARACTERIZATION OF FUNCTIONALLY SEPARABLE AND TIKHONOV SPACES BY MEANS OF DIVERGENT NETS

(Presented by Academician A. N. Tikhonov, 4 VII 1969)

Let \(\mathcal T_1\) be the class of all topological spaces satisfying the first separation axiom; \(\mathcal T_{fs}\) the class of all functionally separable spaces (i.e., topological spaces \(S\) having the property that for any two distinct points \(s^0, s^1 \in S\) there exists a continuous real-valued function \(x\) on \(S\) for which \(x(s^0) \ne x(s^1)\), or, equivalently, there exists a continuous function \(x\) on \(S\) such that \(0 \le x \le 1\), \(x(s^0)=0\), \(x(s^1)=1\)); \(\mathcal T_{cr}\) the class of all completely regular spaces (see \((^1)\), p. 126); \(\mathcal T^n\) (respectively \(\mathcal T^{n'}\)) the class of all topological spaces \(S\) satisfying the following condition: for every divergent net \(\{s_\alpha\}_{\alpha\in A}\) in \(S\) having at least \(n\) (respectively exactly \(n\)) limit points, there exists a bounded continuous real-valued function \(x\), defined on \(S\), such that the net \(\{x(s_\alpha)\}_{\alpha\in A}\) diverges \((n=1,2)\). Further, let
\[ \mathcal T_1^1=\mathcal T_1\cap\mathcal T^1,\quad \mathcal T_1^2=\mathcal T_1\cap\mathcal T^2,\quad \mathcal T_1^{2'}=\mathcal T_1\cap\mathcal T^{2'},\quad \mathcal T_1^{1',2'}=\mathcal T_1\cap\mathcal T^{1'}\cap\mathcal T^{2'},\quad \mathcal T_{1,cr}=\mathcal T_1\cap\mathcal T_{cr}. \]
Spaces of the class \(\mathcal T_{1,cr}\) are called Tikhonov spaces (see \((^1)\), p. 126).

In the present paper the following theorems are proved.

Theorem 1. \(\mathcal T_{fs}=\mathcal T_1^2=\mathcal T_1^{2'}\).

Theorem 2. \(\mathcal T_{1,cr}=\mathcal T_1^1=\mathcal T_1^{1',2'}\).

Theorem 3. Let \(\{S_\beta\}_{\beta\in B}\) be a family of topological spaces, and let \(T\) be their product. Then
\[ 1^\circ.\quad T\in\mathcal T_1^2 \equiv \bigvee_{\beta\in B} S_\beta\in\mathcal T_1^2. \qquad 2^\circ.\quad T\in\mathcal T_1^1 \equiv \bigvee_{\beta\in B} S_\beta\in\mathcal T_1^1. \]

These theorems adjoin the results of the article \((^2)\), in which it was proved that the class of all compact spaces coincides with the class \(\mathcal T_1^0=\mathcal T_1\cap\mathcal T^0\), where \(\mathcal T^0\) is the class of all topological spaces \(S\) having the following property: for every divergent net \(\{s_\alpha\}_{\alpha\in A}\) in \(S\) there exists a bounded continuous real-valued function \(x\), defined on \(S\), such that the net \(\{x(s_\alpha)\}_{\alpha\in A}\) diverges.

Proof of Theorem 1. Taking into account the obvious implication \(S\in\mathcal T_1^2 \Rightarrow S\in\mathcal T_1^{2'}\), it is necessary to establish that: a) \(S\in\mathcal T_{fs}\Rightarrow S\in\mathcal T_1^2\), and b) \(S\in\mathcal T_1^{2'}\Rightarrow S\in\mathcal T_{fs}\).

a) Suppose that \(S\in\mathcal T_{fs}\) and that \(\{s_\alpha\}_{\alpha\in A}\) is a net in \(S\) having at least 2 limit points. Let \(s^0\) and \(s^1\) be any two of them. Take a continuous function \(x\) on \(S\) such that the conditions \(0\le x\le 1\), \(x(s^0)=0\), \(x(s^1)=1\) are satisfied; put
\[ V^0=\{s:\ s\in S,\ x(s)\le 1/3\},\quad V^1=\{s:\ s\in S,\ x(s)\ge 2/3\},\quad A_i=\{\alpha:\ \alpha\in A,\ s_\alpha\in V^i\}\quad (i=0,1). \]
Since \(s^0\) and \(s^1\) are limit points of the net \(\{s_\alpha\}_{\alpha\in A}\), and \(V^0\) and \(V^1\) are neighborhoods respectively of \(s^0\) and \(s^1\), the sets \(A_0\) and \(A_1\) are cofinal with \(A\). Therefore \(\{x(s_\alpha)\}_{\alpha\in A_i}\) \((i=0,1)\) are subnets of the net \(\{x(s_\alpha)\}_{\alpha\in A}\). Hence it follows that the net \(\{x(s_\alpha)\}_{\alpha\in A}\) diverges, for otherwise we would have had

common limit point of all its subnets, which in fact does not exist, since
\(\lim\sup_{\alpha\in A_0} x(s_\alpha) \le 1/3\), while \(\lim\inf_{\alpha\in A_1} x(s_\alpha) \ge 2/3\). Thus it has been shown that \(S \in \mathcal T_{fs} \Rightarrow S \in \mathcal T^2\). But \(S \in \mathcal T_{fs} \Rightarrow S \in \mathcal T_1\); consequently, \(S \in \mathcal T_{fs} \Rightarrow S \in \mathcal T_1^2\).

b) Let \(S \in \mathcal T_1^2\), \(s_1, s_2 \in S\), \(s_1 \ne s_2\). Consider the sequence \(\{s_n\}_{n\in N}\), where \(s_n=s_1\) for odd \(n\) and \(s_n=s_2\) for even \(n\). This sequence has exactly two limit points (\(s_1\) and \(s_2\)) and diverges (taking into account that \(S \in \mathcal T_1\)); hence, by the condition, there exists a bounded real-valued function \(x\), continuous on \(S\), such that the sequence \(\{x(s_n)\}_{n\in N}\) diverges. But this is possible only when \(x(s_1)\ne x(s_2)\). Thus, \(S \in \mathcal T_1^2 \Rightarrow S \in \mathcal T_{fs}\).

For the proof of Theorem 2 the following two lemmas will be needed.

Lemma 1. Let \(S \in \mathcal T_{fs}\), let \(s^0\) be a non-isolated point of the space \(S\), and let \(\{V_\alpha(s^0)\}_{\alpha\in A}\) be the system of all possible neighborhoods of the point \(s^0\) having the form
\[ \{s:\ s\in S,\ |x(s)-x(s^0)|\le \varepsilon\}, \]
where \(x\) is a bounded real-valued function continuous on \(S\), \(\varepsilon>0\). Then the set \(A\), partially ordered by the relation \(\alpha\le \alpha'\), meaning, by definition, that \(V_\alpha(s^0)\supseteq V_{\alpha'}(s^0)\), is directed and has no maximal element.

Proof. Let \(\alpha_1,\alpha_2\in A\),
\[ V_{\alpha_i}(s^0)=\{s:\ s\in S,\ |x_i(s)-x_i(s^0)|\le \varepsilon_i\}\quad (i=1,2). \]
Putting
\[ x_3(s)=\max\bigl(|x_1(s)-x_1(s^0)|,\ |x_2(s)-x_2(s^0)|\bigr),\quad \varepsilon_3=\min(\varepsilon_1,\varepsilon_2), \]
\[ V_{\alpha_3}(s^0)=\{s:\ s\in S,\ x_3(s)\le \varepsilon_3\}, \]
we shall have \(\alpha_i\le \alpha_3\) \((i=1,2)\), which means that the set \(A\) is directed.

Let \(\alpha'\) be an arbitrarily fixed element of \(A\). Choose in the set \(V_{\alpha'}(s^0)\) some point \(s^1\) different from \(s^0\) (such a point exists, since \(s^0\) is a non-isolated point of the space \(S\)). By the condition, there exists a bounded function \(x_0\), continuous on \(S\), such that \(x_0(s^0)=0\), \(x_0(s^1)=1\). Put
\[ V_{\alpha''}(s^0)=\{s:\ s\in S,\ |x_0(s)|\le 1/2\}, \]
and let \(\alpha'''\) be some majorant of the set \(\{\alpha',\alpha''\}\); then \(\alpha'\ne \alpha'''\), for
\[ s^1\in V_{\alpha'}(s^0)\setminus V_{\alpha''}(s^0)\subseteq V_{\alpha'}(s^0)\setminus V_{\alpha'''}(s^0). \]
Since \(\alpha'\) was chosen arbitrarily, this shows that \(A\) has no maximal element.

Lemma 2. Let \(B=A^n\), where \(A\) is a set directed by some relation \(\le\), in which there is no maximal element, and let \(n\) be an arbitrarily fixed natural number. Let \(B_1\) be the diagonal in \(B\), and let \(B_k\) \((k=2,\ldots,n)\) be the set in \(B\) consisting of all possible rows \((a_1,\ldots,a_n)\) such that among the elements \(a_1,\ldots,a_n\) there are exactly \(k\) distinct ones. If \(B\) is partially ordered by the relation
\[ (a_1,\ldots,a_n)\le (a_1',\ldots,a_n'), \]
meaning, by definition, that \(a_i\le a_i'\) \((i=1,\ldots,n)\), then each \(B_k\) \((k=1,\ldots,n)\) will be cofinal with \(B\).

Proof. Let \(\beta=(a_1,\ldots,a_n)\) be an arbitrarily chosen element of the set \(B\); let \(a_1'\) be some majorant of the set \(\{a_1,\ldots,a_n\}\). Since \(\beta_1=(a_1',\ldots,a_1')\in B_1\) and \(\beta\le \beta_1\), this proves that \(B_1\) is cofinal with \(B\). Let \(2\le k\le n\). Since \(A\) has no maximal element, there exist elements \(a_2',\ldots,a_k'\in A\) such that \(a_1',a_2',\ldots,a_k'\) are pairwise distinct and
\[ a_1'\le a_2'\le \cdots \le a_k'. \]
The element
\[ \beta_k=(a_1',a_2',\ldots,a_k',a_k',\ldots,a_k')\in B_k \]
and \(\beta\le \beta_k\). Hence \(B_k\) is cofinal with \(B\) \((k=2,\ldots,n)\).

Proof of Theorem 2. Taking into account the obvious implication
\[ S\in \mathcal T_1^1 \Rightarrow S\in \mathcal T_1^{1',2'}, \]
it is necessary to establish that: a) \(S\in \mathcal T_{1,cr}\Rightarrow S\in \mathcal T_1^1\) and b) \(S\in \mathcal T_1^{1',2'}\Rightarrow S\in \mathcal T_{1,cr}\).

a) Suppose that \(S\in \mathcal T_{cr}\) and that \(\{s_\alpha\}_{\alpha\in A}\) is a divergent net in \(S\) having at least one limit point. Let \(s^0\) be any one of them. Since \(s^0\) is not a point of convergence of the net \(\{s_\alpha\}_{\alpha\in A}\), there exists an open neighborhood \(V^0\) of the point \(s^0\) such that the set
\[ A_0=\{\alpha:\ \alpha\in A,\ s_\alpha\in S\setminus V^0\} \]
is cofinal with \(A\). Choose a function \(x\), continuous on \(S\), so that the following conditions are satisfied:
\[ 0\le x\le 1,\quad x(s^0)=0,\quad x=1 \text{ on } S\setminus V^0, \]
and put
\[ V^1=\{s:\ s\in S,\ x(s)\le 1/2\},\quad A_1=\{\alpha:\ \alpha\in A,\ s_\alpha\in V^1\}. \]
Since \(s^0\) is a limit point of the net \(\{s_\alpha\}_{\alpha\in A}\) and \(V^1\) is a neighborhood of the point \(s^0\), \(A_1\) is cofinal with \(A\). From this we conclude, as

and in the proof of item a) of Theorem 1, that the net \(\{x(s_\alpha)\}_{\alpha\in A}\) diverges (since \(\lim_{\alpha\in A_0} x(s_\alpha)=1\), while \(\limsup_{\alpha\in A_1} x(s_\alpha)\leq 1/2\)). Thus,
\[ S\in \mathscr T_{cr}\Rightarrow S\in \mathscr T'_1, \]
whence
\[ S\in \mathscr T_{1,cr}\Rightarrow S\in \mathscr T'_1. \]

b) To prove the implication
\[ S\in \mathscr T'^{\,2}_1 \Rightarrow S\in \mathscr T_{1,cr}, \]
or, equivalently, the implication
\[ (S\in \mathscr T_{fs})\wedge(S\notin \mathscr T_{cr})\Rightarrow S\in \mathscr T', \tag{*} \]
we first note the following two facts: 1) \(S\in \mathscr T_{cr}\) if and only if, for every point \(s^0\in S\), the system \(\{V_\alpha(s^0)\}_{\alpha\in A}\) (see Lemma 1) is a base of neighborhoods of the point \(s^0\); 2) if \(S\in \mathscr T'_1\) and \(s^0\) is an isolated point of the space \(S\), then the function \(x\), defined on \(S\) by the equalities \(x(s^0)=0\), \(x(s)=1\) for \(s\ne s^0\), is continuous on \(S\), and therefore
\[ \{s^0\}=\{s:\ s\in S,\ x(s)\leq 1/2\}\in \{V_\alpha(s^0)\}_{\alpha\in A}. \]

Proceeding to the proof of the implication \((*)\), suppose that
\[ (S\in \mathscr T_{fs})\wedge(S\notin \mathscr T_{cr}); \]
then (by virtue of the condition \(S\notin \mathscr T_{cr}\)) in \(S\) there is a point \(s^0\) and a neighborhood \(V\) of this point such that
\[ V_\alpha(s^0)\cap (S\setminus V)\ne \varnothing \]
for every \(\alpha\in A\). Hence, and from the condition \(S\in \mathscr T_{fs}\) (which entails \(S\in \mathscr T_1\)), it follows that \(s^0\) is a non-isolated point of the space \(S\). Put \(B=A^2\) and partially order \(A\) and \(B\) as was done respectively in Lemma 1 and in Lemma 2 for \(n=2\). Then \(A\) will have no maximal element (Lemma 1), and each \(B_k\) \((k=1,2)\) will be a cofinal part of \(B\) (Lemma 2). We construct in \(S\) a net \(\{s_\beta\}_{\beta\in B}\) as follows: if \(\beta=(a,a)\in B_1\), then as \(s_\beta\) we take some element of \(V_a(s^0)\cap(S\setminus V)\); if \(\beta\in B_2\), then we put \(s_\beta=s^0\). Since \(B_2\) is cofinal in \(B\), \(s^0\) is a cluster point of the net \(\{s_\beta\}_{\beta\in B}\). However, \(s^0\) is not a point of convergence of the net \(\{s_\beta\}_{\beta\in B}\), for \(V\) is a neighborhood of the point \(s^0\),
\[ B_1=\{\beta:\ \beta\in B,\ s_\beta\notin V\}, \]
and \(B_1\) is cofinal in \(B\). We shall show that any point \(s^1\) distinct from \(s^0\) cannot be a cluster point of the net \(\{s_\beta\}_{\beta\in B}\). Indeed, the condition \(S\in \mathscr T_{fs}\) means the existence of a continuous function \(x\) on \(S\) such that \(x(s^0)=0\), \(x(s^1)=1\). Putting
\[ V^0=\{s:\ s\in S,\ |x(s)|\leq 1/3\},\qquad V^1=\{s:\ s\in S,\ |x(s)|\geq 2/3\}, \]
we shall have \(V^0=V_{\alpha_0}(s^0)\) for some \(\alpha_0\in A\), and \(V_{\alpha_0}(s^0)\cap V^1=\varnothing\). It follows that \(s_\beta\notin V^1\) for \(\beta\geq \beta_0=(\alpha_0,\alpha_0)\). This is precisely to say that \(s^1\) is not a cluster point of the net \(\{s_\beta\}_{\beta\in B}\) (since \(V^1\) is a neighborhood of the point \(s^1\)). Thus, the net \(\{s_\beta\}_{\beta\in B}\) diverges and has exactly one cluster point, \(s^0\). If we verify that for every continuous function \(x\) on \(S\) the net \(\{x(s_\beta)\}_{\beta\in B}\) converges, then thereby the required result will be established:
\[ S\notin \mathscr T'. \]
We shall show that the net \(\{x(s_\beta)\}_{\beta\in B}\) converges to \(x(s^0)\). Choose a number \(\varepsilon>0\) and consider the neighborhood
\[ V_{\alpha_\varepsilon}(s^0)=\{s:\ s\in S,\ |x(s)-x(s^0)|\leq \varepsilon\} \]
of the point \(s^0\). Obviously \(s_\beta\in V_{\alpha_\varepsilon}(s^0)\) for
\[ \beta\geq \beta_\varepsilon=(\alpha_\varepsilon,\alpha_\varepsilon), \]
and hence
\[ |x(s_\beta)-x(s^0)|\leq \varepsilon \]
for \(\beta\geq \beta_\varepsilon\). This proves, in view of the arbitrariness of \(\varepsilon\), that
\[ \lim_{\beta\in B} x(s_\beta)=x(s^0). \]

Proof of item \(1^0\) of Theorem 3. Taking Theorem 1 into account, it is necessary to show that
\[ T\in \mathscr T_{fs}\ \equiv\ \bigvee_{\beta\in B} S_\beta\in \mathscr T_{fs}. \]

a) Let \(T\in \mathscr T_{fs}\), \(\beta_0\in B\), \(s_{\beta_0}^1,s_{\beta_0}^2\in S_{\beta_0}\), \(s_{\beta_0}^1\ne s_{\beta_0}^2\). Choose in each \(S_\beta\), for \(\beta\ne\beta_0\), an element \(s_\beta^1\), and define in \(T\) elements \(t^1,t^2\), putting
\[ t^1(\beta_0)=s_{\beta_0}^1,\qquad t^2(\beta_0)=s_{\beta_0}^2,\qquad t^1(\beta)=t^2(\beta)=s_\beta^1 \]
for \(\beta\ne\beta_0\). Since \(t^1\ne t^2\), by the condition there exists a function \(y\), continuous on \(T\), such that
\[ y(t^1)\ne y(t^2). \]
Let
\[ T_0=\{t:\ t\in T,\ t(\beta)=s_\beta^1\ \text{for}\ \beta\ne\beta_0\}, \]
and let \(f:T_0\to S_{\beta_0}\) be the mapping defined by the equality
\[ f(t)=t(\beta_0) \]
for every \(t\in T_0\). This is a one-to-one and bicontinuous mapping of \(T_0\) onto \(S_{\beta_0}\). Put
\[ x(s_{\beta_0})=y(f^{-1}(s_{\beta_0})) \]
for each \(s_{\beta_0}\in S_{\beta_0}\). Obviously, the function \(x\) is continuous on \(S_{\beta_0}\), and moreover
\[ x(s_{\beta_0}^1)=y(f^{-1}(s_{\beta_0}^1))=y(t^1)\ne y(t^2)=y(f^{-1}(s_{\beta_0}^2))=x(s_{\beta_0}^2). \]
Since \(\beta_0\) is an arbitrary element of \(B\), the implication
\[ T\in \mathscr T_{fs}\Rightarrow \bigvee_{\beta\in B} S_\beta\in \mathscr T_{fs} \]
is proved.

b) Let
\[ \bigvee_{\beta\in B} S_\beta\in \mathscr T_{fs},\qquad t^1,t^2\in T,\qquad t^1\ne t^2. \]
Then \(t^1(\beta_0)\ne t^2(\beta_0)\) at least ...

at least for one \(\beta_0 \in B\) there exists a function \(x\), continuous on \(S_{\beta_0}\), such that \(x(t^1(\beta_0)) \ne x(t^2(\beta_0))\). Put \(y(t)=x(t(\beta_0))\) for any \(t \in T\). It is obvious that the function \(y\) is continuous on \(T\), and \(y(t^1)\ne y(t^2)\). This proves the implication

\[ \forall_{\beta\in B} S_\beta \in \mathscr{T}_{fs} \Rightarrow T \in \mathscr{T}_{fs}. \]

Proof of item \(2^\circ\) of Theorem 3 differs only insignificantly from the proof of the assertion

\[ T \in \mathscr{T}^{0} \equiv \forall_{\beta\in B} S_\beta \in \mathscr{T}^{0}, \]

given in article \((^2)\), and therefore it is omitted.

Latvian State University
named after P. Stučka
Riga

Received
30 VIII 1968

REFERENCES

\(^1\) K. Kuratowski, Topology, 1, Moscow, 1966.
\(^2\) M. A. Goldman, Latvian Math. Yearbook, 8, 1970.