Strong universality for totally bounded subsets. Applications to the spaces \(C_ p(X)\). (Universalité forte pour les sous-ensembles totalement bornés. Applications aux espaces \(C_ p(X)\).) (French) Zbl 0882.57014
An important question considered in this work is the following: Are two locally convex metrizable topological vector spaces (from now on called locally convex spaces) homeomorphic if each is homeomorphic to a closed subspace of the other? Such spaces are not necessarily completely metrizable, so the method of absorbers of M. Bestvina and J. Mogilski [Mich. Math. J. 33, 291-313 (1986; Zbl 0629.54011)] is a powerful tool in helping to get answers in this vein. The authors consider the special case when the two spaces are \(Z_\sigma\)’s, i.e., are countable unions of \(Z\)-sets.
In absorber theory one always deals with a certain class \(\mathcal C\) of spaces where \(\mathcal C\) is topological and hereditary for closed subspaces. Starting with a locally convex space \(E\), put \(\mathcal C=\mathcal C(E)\) equal the class of spaces which are homeomorphic to totally bounded closed subspaces of \(E\). Then, \(\sigma\)-\(\mathcal C\) denotes the class of all spaces which are countable unions of elements of \(\mathcal C\).
The authors are able to generalize a result of T. Dobrowolski [Trans. Am. Math. Soc. 313, No. 2, 753-784 (1989; Zbl 0692.57007)] by showing that if \(E\) is infinite-dimensional, then
(a) \(E\) is strongly \(\mathcal C\)-universal;
(b) if in addition \(E\) is a \(Z_\sigma\), then \(E\) is strongly \(\sigma\)-\(\mathcal C\) universal.
One corollary to this result is that if \(E_1\), \(E_2\) are locally convex spaces which are of type \(Z_\sigma\), \(E_1\in\sigma\)-\(\mathcal C(E_2)\), and \(E_2\in\sigma\)-\(\mathcal C(E_1)\), then \(E_1\), \(E_2\) are homeomorphic. This condition, however, is not necessary. But another corollary shows that if \(E_1\), \(E_2\) are \(\sigma\)-precompact, then \(E_1\cong E_2\) if and only if each is homeomorphic to a closed subspace of the other. Additional results in this area are also obtained.
Another part of the paper deals with \(C_p(X)\), the space of maps of \(X\) to \(\mathbb{R}\) with the pointwise topology, and its subspace \(C_p^*(X)\) consisting of all the bounded maps. Always \(X\) is taken to be a regular, countable, non-discrete space. One may find some background in [R. Cauty, T. Dobrowolski and W. Marciszewski, Fundam. Math. 142, No. 3, 269-301 (1993; Zbl 0813.54009)].
Here it is proved that \(C_p(X)\cong C_p^*(X)\) if and only if \(C_p(X)\) is a \(Z_\sigma\). It follows that if \(C_p(X)\) is analytic, then \(C_p(X)\cong C_p^*(X)\). Another theorem states that \(C_p^*(X)\cong C_p(X)\times\sigma\). The space \(\sigma\subset\mathbb{R}^\omega\) consists of all points having only finitely many non-zero coordinates. Other corollaries to these results are also stated.
In absorber theory one always deals with a certain class \(\mathcal C\) of spaces where \(\mathcal C\) is topological and hereditary for closed subspaces. Starting with a locally convex space \(E\), put \(\mathcal C=\mathcal C(E)\) equal the class of spaces which are homeomorphic to totally bounded closed subspaces of \(E\). Then, \(\sigma\)-\(\mathcal C\) denotes the class of all spaces which are countable unions of elements of \(\mathcal C\).
The authors are able to generalize a result of T. Dobrowolski [Trans. Am. Math. Soc. 313, No. 2, 753-784 (1989; Zbl 0692.57007)] by showing that if \(E\) is infinite-dimensional, then
(a) \(E\) is strongly \(\mathcal C\)-universal;
(b) if in addition \(E\) is a \(Z_\sigma\), then \(E\) is strongly \(\sigma\)-\(\mathcal C\) universal.
One corollary to this result is that if \(E_1\), \(E_2\) are locally convex spaces which are of type \(Z_\sigma\), \(E_1\in\sigma\)-\(\mathcal C(E_2)\), and \(E_2\in\sigma\)-\(\mathcal C(E_1)\), then \(E_1\), \(E_2\) are homeomorphic. This condition, however, is not necessary. But another corollary shows that if \(E_1\), \(E_2\) are \(\sigma\)-precompact, then \(E_1\cong E_2\) if and only if each is homeomorphic to a closed subspace of the other. Additional results in this area are also obtained.
Another part of the paper deals with \(C_p(X)\), the space of maps of \(X\) to \(\mathbb{R}\) with the pointwise topology, and its subspace \(C_p^*(X)\) consisting of all the bounded maps. Always \(X\) is taken to be a regular, countable, non-discrete space. One may find some background in [R. Cauty, T. Dobrowolski and W. Marciszewski, Fundam. Math. 142, No. 3, 269-301 (1993; Zbl 0813.54009)].
Here it is proved that \(C_p(X)\cong C_p^*(X)\) if and only if \(C_p(X)\) is a \(Z_\sigma\). It follows that if \(C_p(X)\) is analytic, then \(C_p(X)\cong C_p^*(X)\). Another theorem states that \(C_p^*(X)\cong C_p(X)\times\sigma\). The space \(\sigma\subset\mathbb{R}^\omega\) consists of all points having only finitely many non-zero coordinates. Other corollaries to these results are also stated.
Reviewer: L.R.Rubin (Norman)
