\(G_ \delta\)-open functionally bounded subsets in topological groups
For a topological space \(X\), let \(C(X)\) denote the ring of all continuous real-valued functions on \(X\). A subset \(A\) of \(X\) is called functionally bounded (or relatively pseudocompact) in \(X\) if each \(f\in C(X)\) is bounded on \(A\), and \(A\) is said to be hyperbounded in \(X\) when \(\inf\{f(x)\): \(x\in A\}>0\) for every \(f\in C(X)\) which is positive on \(A\) [cf. \textit{T. Isiwata}, Pac. J. Math. 20, 455-480 (1967; Zbl 0149.405)]. A subset \(A\) of \(X\) is said to be \(G_ \delta\)-open in \(X\) when for any point \(x\in A\), there is a \(G_ \delta\)-set \(U\) in \(X\) such that \(x\in U\subset A\), and call \(A\) \(G_ \delta\)-dense in \(X\) if every \(G_ \delta\)-set in \(X\) intersects \(A\). This paper deals with functionally bounded sets of a topological group which are \(G_ \delta\)-dense in their closure in the bilateral completion \({\mathcal G}\) of the group. The following main results are proved: Theorem 1. Suppose that \(A\) is a functionally bounded subset of a topological group \(G\). Then \(A\) is hyperbounded if and only if \(A\) is \(G_ \delta\)-dense in \(cl_{\mathcal G} A\). -- Theorem 2. If \(A\) is a functionally bounded \(G_ \delta\)-open subset of a topological group \(G\), then the Cartesian product of \(cl_ G A\) by any pseudocompact space is a pseudocompact space. -- Theorem 3. Let \(G\) be a topological group and let \(B\) be a functionally bounded set in \(G\) which contains a \(G_ \delta\)-open subset \(A\) in \(B\). Then \(B\) is hyperbounded \(\Leftrightarrow B\) is pseudocompact \(\Leftrightarrow B\) is \(C\)-embedded in \(G\Leftrightarrow B\) is \(C\)-embedded in \(cl_{\mathcal G} B\Leftrightarrow B\) is \(P\)-embedded in \(G\Leftrightarrow B\) is \(P\)-embedded in \(cl_{\mathcal G} B\).
scientific article; zbMATH DE number 500679
scientific article