On function spaces of Corson-compact spaces.

*(English)*Zbl 0835.54016First some definitions. A space is Corson-compact if it is a compact subset of a \(\sigma\)-product (countable support) of the reals. Corson- compacts were introduced because they nicely generalize the Eberlein- compacts which are the subspaces of Banach spaces compact in the weak-* topology. Given a space \(X\), \(C_p (X)\) denotes the space of continuous real-valued functions with pointwise convergence. Finally a Lindelöf \(\Sigma\)-space is one which is the continuous image of a Lindelöf space which itself maps perfectly onto a metric space.

The author proves three very nice results about Corson-compact spaces and their function spaces. However, for most readers, what will stand out the most about this article is the approach that the author has developed to this study. The first result is a generalization of Gul’ko’s result that if \(X\) is a Lindelöf \(\Sigma\)-space, then each compact subspace of \(C_p (X)\) is Corson. The third result answers a question of Arkhangel’skij and genralizes a result of R. Pol by showing that \(X\) is Corson if \(C_p (X)\) (for compact \(X\)) is a continuous image of a closed subset of some closed subset of \(Z \times (L_\tau)^\omega)\) (where \(Z\) is compact and \(L_\tau\) is the one-point Lindelöf extension of the discrete space of cardinality \(\tau\)).

As mentioned, what is most striking about the paper is the approach – an essential and natural use of countable elementary submodels. The elegance of the proofs of the three mentioned results give ample testimony that this approach can be used very profitably in this context. Another such example is in the author’s paper [Commentat. Math. Univ. Carol. 32, No. 3, 545-550 (1991; Zbl 0769.54025)]. More specifically, the class \(\Omega\) is defined using elementary submodels and the second main result of the paper is that a compact space is Corson if and only if \(C_p (X)\) is in the class \(\Omega\). The author develops a general framework for the utilization of elementary submodels in the study of topology. Given a space \(X\) which is a member of a countable model \(M\), he shows there is a metric space \(X (M)\) and a canonical mapping from \(X\) onto \(X(M)\) (which is, for example, one-to-one on \(X \cap M\)). A space is said to be in \(\Omega\), if the closure of \(X \cap M\) maps onto \(X (M)\) (for all suitably elementary \(M\)). A stationary, co-stationary subset, \(S\), of \(\omega_1\) would be a simple example of a space which is not in \(\Omega\) (if \(M \cap S\) is not in \(S\), then \(M \cap S\) is closed in \(S\) and you will find that the points outside of \(M\) all map to the same point not in the image of \(M \cap S\)).

The author proves three very nice results about Corson-compact spaces and their function spaces. However, for most readers, what will stand out the most about this article is the approach that the author has developed to this study. The first result is a generalization of Gul’ko’s result that if \(X\) is a Lindelöf \(\Sigma\)-space, then each compact subspace of \(C_p (X)\) is Corson. The third result answers a question of Arkhangel’skij and genralizes a result of R. Pol by showing that \(X\) is Corson if \(C_p (X)\) (for compact \(X\)) is a continuous image of a closed subset of some closed subset of \(Z \times (L_\tau)^\omega)\) (where \(Z\) is compact and \(L_\tau\) is the one-point Lindelöf extension of the discrete space of cardinality \(\tau\)).

As mentioned, what is most striking about the paper is the approach – an essential and natural use of countable elementary submodels. The elegance of the proofs of the three mentioned results give ample testimony that this approach can be used very profitably in this context. Another such example is in the author’s paper [Commentat. Math. Univ. Carol. 32, No. 3, 545-550 (1991; Zbl 0769.54025)]. More specifically, the class \(\Omega\) is defined using elementary submodels and the second main result of the paper is that a compact space is Corson if and only if \(C_p (X)\) is in the class \(\Omega\). The author develops a general framework for the utilization of elementary submodels in the study of topology. Given a space \(X\) which is a member of a countable model \(M\), he shows there is a metric space \(X (M)\) and a canonical mapping from \(X\) onto \(X(M)\) (which is, for example, one-to-one on \(X \cap M\)). A space is said to be in \(\Omega\), if the closure of \(X \cap M\) maps onto \(X (M)\) (for all suitably elementary \(M\)). A stationary, co-stationary subset, \(S\), of \(\omega_1\) would be a simple example of a space which is not in \(\Omega\) (if \(M \cap S\) is not in \(S\), then \(M \cap S\) is closed in \(S\) and you will find that the points outside of \(M\) all map to the same point not in the image of \(M \cap S\)).

Reviewer: A.Dow (North York)

##### MSC:

54C35 | Function spaces in general topology |