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Session 36. Topology in Functional Analysis |
On compact subspaces of the space of separately continuous functions with the cross-uniform topology |
| Oleksandr V. Maslyuchenko, Jurij Fedkovych Chernivtsi National University, Ukraine |
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In [1] the authors proposed a natural topologization of the space \(S\) of all separately continuous functions \(f:[0,1]^2\to \mathbb{R}\) called
the topology of the sectionally uniform convergence. This topology can be considered on the space \(S(X\times Y)\) of all separately continuous functions \({f:X\times Y\to \mathbb{R}}\) for any topological spaces \(X\) and \(Y\). A base of this topology is given by the sets
\({
W_{E,\varepsilon}(f_0)=\{f\in S(X\times Y):\forall p\in\mathrm{cr}E\quad |f(p)-f_0(p)|<\varepsilon\},
}\)
where \(E\) is a finite subset of \(X\times Y\), \(\varepsilon>0\), \(f_0\in S(X\times Y)\) and \(\mathrm{cr}E=(X\times\mathrm{pr}_Y(E))\cup(\mathrm{pr}_X(E)\times Y)\) is
the cross of the set \(E\). We call this topology the cross-uniform topology
and always endow the space \(S(X\times Y)\) by this topology. If \(X\) and \(Y\) are compacta then \(S(X\times Y)\) is a topological vector space.
In [1] it was proved only that \(S=S([0,1]^2)\) is a separable non-metrizable complete topological vector space, and the authors asked about the other properties of \(S\).
The following result was obtained in collaboration with the authors of [1].
Theorem: Let \(X,Y\) be compacta without isolated points. Then \(S(X\times Y)\) is meager and barreled. Another intrigued question on the space \(S(X\times Y)\) is the problem on description of compact subspaces of \(S(X\times Y)\) for any compacta \(X\) and \(Y\). Compact subspaces of \(B_1(X)\) (= the space of all Baire one function with the pointwise topology) are, so-called, Rosenthal compacta if \(X\) is a Polish space. Since \(S([0,1]^2)\subseteq B_1([0,1]^2)\), we expected the appearance of some Rosenthal type compacta. But it turns out that the structure of compact subspaces of \(S(X\times Y)\) is simpler. Let \(w(X)\) denote the weight of a topological space \(X\) and let \(c(X)\) denote the cellularity of \(X\). Theorem: Let \(X,Y\) be infinity compacta and \(K\) be a compact. Then \(K\) embeds into \(S(X\times Y)\) if and only if
\({w(K)\le\min\{c(X),c(Y)\}}\).
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References
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