An example of a topological group
A topological space \(X\) is a \(P\)-\textit{space} if for every family \(G_n\subset X\) (\(n\in \mathbb{N}\)) of open subsets of \(X\), the intersection \(\cap_{n\in \mathbb{N}} G_n\) is an open subset of \(X\). A topological group \(G\) is called \textit{o-bounded} if for every sequence \((U_n)_{n\in \mathbb{N}}\) of neighborhoods of the identity in \(G\) there is a sequence \((K_n)_{n\in \mathbb{N}}\) of finite subsets of \(G\) such that \(G=\bigcup_{n\in \mathbb{N}}U_n K_n\). For a given topological group \(G\) one defines a game where the first player chooses a neighborhood \(U_n\) of the identity in \(G\), and the second player answers with a finite set \(K_{n}\subset G\). The second player wins if \(G=\bigcup_{n\in \mathbb{N}}U_n K_n\). A topological group is called \textit{strictly o-bounded} if there is a winning strategy for the second player. In this paper the authors provide an example of a topological group \(G\) which is a Lindelöf \(P\)-space but is not strictly bounded (this is also an example of a group which is o-bounded but not strictly o-bounded). Thus, the paper clarifies the relation existent among the notions defined above and answers some questions of \textit{C. Hernández} [Topology Appl. 102, 101-111 (2000; Zbl 0942.22001)] and \textit{M. Tkachenko} [ibid. 86, 179-231 (1998; Zbl 0955.54013)].
scientific article; zbMATH DE number 1901908