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Session 39. Contributed talks |
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
\({w(K)\le\min\{c(X),c(Y)\}}\). |
References
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