MATHEMATICS

V. TRNKOVA

ON UNIONS OF STRONGLY PARACOMPACT SPACES

(Presented by Academician P. S. Aleksandrov on 30 III 1962)

1. It is known that if a topological space has a locally finite open covering consisting of sets with paracompact closures, then it is itself paracompact. An analogous assertion is true also for many other properties. The following proposition holds (in which a class of spaces is called \(F\)-hereditary if it contains, together with every space, all its closed subspaces):

Proposition 1. Let \(V\) be an \(F\)-hereditary class of normal spaces and suppose: a) if \(P\) is a union of disjoint open subspaces from the class \(V\), then \(P \in V\); b) if \(P = G_1 \cup G_2\), the \(G_i\) are open, \(\overline{G_i} \in V\), then \(P \in V\). Let the space \(S\) have a locally finite open covering consisting of spaces with closures belonging to \(V\). Then \(S\) belongs to the class \(V\).

Proof. Let there be a covering \(G\) with the indicated properties. Insert into it a locally finite open covering of the form \(\bigcup_{n=1}^{\infty} A_n\), where the \(A_n\) are discrete systems. Denote by \(H_n\) the union of the sets \(A \in A_n\); obviously, \(\overline{H_n} \in V\). Insert into the countable locally finite covering \(\{H_n\}\) a star-finite open covering (see (2)). Each component \(U\) of this covering* has the form
\[ U=\bigcup_{k=1}^{\infty} U_k, \]
where the \(U_k\) are open, \(\overline{U_k} \in V\), \(\overline{U_k} \subset U_{k+1}\). Put
\[ G_k=\overline{U_{k+1}}-U_k,\qquad M=\bigcup_{k=1}^{\infty} G_{2k},\qquad N=\bigcup_{k=1}^{\infty} G_{2k+1}. \]
Then \(M\), \(N\), and \(U=M\cup N\) belong to the class \(V\), whence it follows that \(S\in V\).

Remark. If the class \(V\) satisfies conditions a), b), then the class of all spaces hereditarily belonging to \(V\) also satisfies them.

1. The class of strongly paracompact spaces satisfies condition a) of Proposition 1, but not condition b); as an example one may take the product of an open interval and the “point hedgehog”*.

We shall now construct, for \(n=2,3,\ldots\), a metric space \(P\) such that:

1) \(P\) is a union of \(n\), but not of \(n-1\), open subsets with strongly paracompact closures;

2) \(P\) is a union of two strongly paracompact closed subsets.

Let \(S\) be an \((n-1)\)-dimensional cube lying in the plane \(x_n=0\) of the space \(E_n\). By known theorems there exists \(\varepsilon>0\) such that if \(M_i\)

* By a component \(U\) of a covering \(\mathcal U\) we mean the union of the sets of a maximal subsystem \(\mathcal U'\subset \mathcal U\) such that for any \(G,H\in \mathcal U'\) there exist \(G_1,\ldots,G_k\in \mathcal U\) with \(G\cap G_1\ne\varnothing\), \(G_i\cap G_{i+1}\ne\varnothing\) \((i=1,\ldots,k-1)\), \(G_k\cap H\ne\varnothing\).

** A Hausdorff space is called strongly paracompact if into every open covering of it one can insert a star-finite open covering.

*** By a “point hedgehog” we mean a metrizable space all of whose points are isolated except one, every neighborhood of which is uncountable.

open in \(S\), \(S=\bigcup_{i=1}^{n-1} M_i\), then at least one of the components of the sets \(M_i\) has diameter \(>\varepsilon\). Denote by \(\mathcal Y\) the interval \(\langle -1,1\rangle\), and by \(\Phi\) the set of all its homeomorphic images \(f\) in \(S\) such that the diameter of \(f(\mathcal Y)\) is not less than \(\varepsilon\). Take some abstract set \(A\) and its mapping \(\varphi\) onto \(\Phi\) such that all \(\varphi^{-1}(f)\) are uncountable; instead of \(\varphi(a)\) we shall write \(\varphi_a\). For \(a\in A\) denote by \(T_a\) the set of all points \([x_1,\ldots,x_n,a]\in E_n\times A\) such that
\([x_1,\ldots,x_{n-1},0]=\varphi_a\left(\sin {1\over x_n}\right)\); put \(C_a=\varphi_a(\mathcal Y)\cup T_a\), and denote by \(\sigma_a\) the metric on \(C_a\) obtained by means of the obvious embedding in \(E_n\). We now introduce a metric \(\rho\) on \(P=\bigcup_{a\in A} C_a\), setting \(\rho(x,y)=\sigma_a(x,y)\), if \(x,y\) belong to one and the same \(C_a\); \(\rho(x,y)=\inf(\sigma_a(x,z)+\sigma_\beta(z,y))\), if \(x\in C_a,\ y\in C_\beta,\ a\ne\beta\).

Denote by \(M\) and, respectively, by \(Q\) the subset of the space \(P\) consisting of all \(x\in S\) and all points of the form \([x_1,\ldots,x_n,a]\in T_a\) satisfying the condition \(\sin {1\over x_n}\ge -{1\over 2}\) (respectively, \(\sin {1\over x_n}\le {1\over 2}\)). Obviously, \(P=M\cup Q\), and \(M,Q\) are closed; it is easy to establish that \(M,Q\) are strongly paracompact.

The space \(P\) is not the union of \(n-1\) open subsets: if \(P=\bigcup_{i=1}^{n-1}G_i\) with \(G_i\) open, then at least one of the sets \(G_i\cap S\) has a component of diameter \(>\varepsilon\), and therefore contains some \(f(\mathcal Y)\), \(f\in\Phi\); then, for a suitable \(\delta>0\), the set consisting of all points \(f(\mathcal Y)\) and all \([x_1,\ldots,x_n,a]\in T\) such that \(\varphi_a=f,\ 0\le x_n\le\delta\), is contained in \(G_i\), is closed in \(P\), but is not strongly paracompact.

Finally, let us show that \(P\) is the union of \(n\) open subsets with strongly paracompact closures. From known theorems it follows that \(S=\bigcup_{i=1}^{n}H_i\), where \(H_i\) are open in \(S\) and their components have diameter \(<\varepsilon\). Denote by \(Q_i\) the set consisting of all \(x\in H_i\) and all \([x_1,\ldots,x_n,a]\) such that \([x_1,\ldots,x_{n-1},0]\in H_i\). Then the \(H_i\) are open and their closures are strongly paracompact.

1. It is easy to verify the following propositions:

Proposition 2. Let the space \(P\) be the union of a locally finite system of order \(\le n\) of open subsets with strongly paracompact closures. Then \(P\) is the union of \(n+1\) such subsets.

Proposition 3. Let the space \(P\) be the union of a locally finite system of open subsets with strongly paracompact closures. Then \(P\) is the union of a locally finite countable system of such sets.

Proposition 4. A space \(P\) is paracompact and locally strongly paracompact if and only if it is the union of a countable locally finite system of its open subsets with strongly paracompact closures.

1. As is known (see also the example given above), a space that is the union of two closed strongly paracompact subspaces need not be strongly paracompact. It may even fail to be locally strongly paracompact, as is shown by example 3 from the article \((^3)\) (to which Yu. M. Smirnov drew the author’s attention). However, the following assertion holds:

Proposition 5. If \(P=M_1\cup M_2\), \(M_1,M_2\) are closed and strongly paracompact, and the space \(M_1\cap M_2\) is locally Lindelöf*, then \(P\) is strongly paracompact.

* A space is called Lindelöf if from each of its open coverings one can choose a countable one.

Proof. Put \(M_1=A_1,\ \overline{P-M_1}=A_2,\ Q=A_1\cap A_2\). Then \(Q\) is a locally Lindelöf paracompact space, and therefore there exists a discrete system
\(G=\{G_\lambda;\lambda\in\Lambda\}\) of open-and-closed subsets of \(Q\) such that
\(\bigcup_{\lambda\in\Lambda}G_\lambda=Q\) and each \(G_\lambda\) is a Lindelöf space \((^5)\). Let \(\mathcal W\) be an open covering of \(P\); it is enough to inscribe in it a star-countable open covering \((^3)\). \(P\) is paracompact, and therefore there exists an open covering \(\mathcal U\) of it such that no \(U\in\mathcal U\) meets two sets \(G_\lambda\) and the system \(\{\operatorname{st} U;\ U\in\mathcal U\}^*\) is inscribed in \(\mathcal W\). Put
\(\mathcal U_i=\{U\cap A_i;\ U\in\mathcal W\},\ i=1,2\). Let \(\mathcal V_i\) be a star-countable open (in \(A_i\)) covering of the space \(A_i\), inscribed in \(\mathcal U_i\). Let \(\mathcal V_{\lambda,i}\) be a countable subsystem of \(\mathcal V_i\) which covers \(G_\lambda\). If now we put

\[ \mathcal V_\lambda=\{\operatorname{Int}(V_1\cup V_2);\ V_1\in\mathcal V_{\lambda,1},\ V_2\in\mathcal V_{\lambda,2}\}, \]

then it is clear that \(\mathcal V_\lambda\) is a countable system of open subsets of \(P\) covering \(G_\lambda\). If, moreover, we put

\[ \mathcal X_i=\{V-Q;\ V\in\mathcal V_i-\bigcup_{\lambda\in\Lambda}\mathcal V_{\lambda,i}\},\ i=1,2;\qquad \mathcal V=\mathcal X_1\cup\mathcal X_2\cup\bigcup_{\lambda\in\Lambda}\mathcal V_\lambda, \]

then, obviously, \(\mathcal V\) is a star-countable open covering of \(P\), inscribed in \(\mathcal W\).

1. We now give an answer to the problem of the additivity of the class of strongly paracompact spaces with respect to open subsets (the author’s attention was also drawn to this problem by Yu. M. Smirnov), namely, we shall construct a paracompact but not strongly paracompact space \(P\) and its open strongly paracompact subspaces \(G_1, G_2\), such that \(P=G_1\cup G_2\). Let \(T_{\omega_1}\) be the set of all countable ordinal numbers. Let \(M=T_{\omega_1}\times\langle 0,1\rangle\). On \(M\) we take the lexicographic order \(<\) and the topology determined by this order. Also denote
   \(C=\{[x,0];\ x\in T_{\omega_1}\}\),
   \(D=\{[x,\tfrac12];\ x\in T_{\omega_1}\}\). Let \(A\) be an uncountable discrete space,
   \(Q=(\xi)\cup(M\times A)\); on \(Q\) we take the following topology: on \(M\times A\) we take the product topology of the spaces; a complete system of neighborhoods of the point \(\xi\) will be the system \(\{U_x;\ x\in M\}\), where
   \(U_x=\{[m,a]\in M\times A:\ x<m\}\cup(\xi)\).

Put also

\[ S_1=(\xi)\cup[(M-C)\times A],\qquad S_2=(\xi)\cup[(M-D)\times A], \]

\[ S_3=(\xi)\cup[(M-C\cup D)\times A]. \]

Put \(P^*=Q\times\langle 0,1\rangle\) and, as the space \(P\) with the required properties, take the following subspace of it:
\[ P=(S_1\times\{0\})\cup(S_3\times(0,1))\cup(S_2\times\{1\}). \]
It can be verified that \(P\) is not strongly paracompact, whereas the sets
\[ G_1=(S_1\times\{0\})\cup(S_3\times(0,1)) \]
and
\[ G_2=(S_3\times(0,1))\cup(S_2\times\{1\}) \]
are strongly paracompact.

Karlov University
Prague, Czechoslovakia

Received
28 III 1962

References

1. M. Katetov, Czechoslovak Mathematical Journal, 2 (77), 333 (1952).
2. K. Morita, Math. Japon., 1, 60 (1948).
3. Yu. M. Smirnov, Izv. Akad. Nauk SSSR, Ser. Mat., 20, 253 (1956).
4. Yu. M. Smirnov, Fund. Math., 43, 387 (1956).
5. V. Šedivá, Czechoslovak Mathematical Journal, 9 (84), 50 (1959).

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* \(\operatorname{st} U\) denotes the union of all sets of the covering \(\mathcal U\) having nonempty intersection with \(U\).