The maximal \(G\)-compactifications of \(G\)-spaces with special actions (Q2782203)

All spaces here are Tikhonov and all mappings are continuous. Let \(G\) be a topological group. A \(G\)-space \(X\) (phase space) is any space with a continuous action of \(G\). A compact \(G\)-space \(bX\) is a \(G\)-compactification of a \(G\)-space \(X\) if there is an equivariant dense embedding of \(X\) into \(bX\). If a \(G\)-space \(X\) has a \(G\)-compactification, then it has a largest \(G\)-compactification which is denoted by \(\beta_GX\). The authors investigate the compactification \(\beta_GX\). For example, let \(\mathbb{A}\) denote the family of all open neighborhoods of the identity in \(G\). For each \(O\in \mathbb{A}\), let \(\gamma_O= \{Ox: x\in X\}\) and \(\overline{\gamma}_O= \{\text{cl}(Ox): x\in X\}\). Denote by \(U_G\) \((\overline{U}_G)\) the family of all coverings of \(X\) which have a refinement of the form \(\gamma_O\) \((\overline{\gamma}_O)\) for some \(O\in \mathbb{A}\). The family \(U_G\) is a uniformity on \(X\) which is not necessarily compatible with the topology on \(X\). Next, let \( U^*\) be the totally bounded uniformity on \(X\) compatible with its topology such that any bounded continuous function on \(X\) is uniformly continuous with respect to it. The authors show that if \(X\) is a \(G\)-space with the property that \(U_G\) is finer than \(U^*\), then \(\beta_GX= \beta X\). They also show that if \(\overline{U}_G\) is a uniformity compatible with the topology of the \(G\)-space \(X\), then \(\beta_GX\) is the Samuel compactification of \(X\) with respect to \(\overline{U}_G\). In addition to other results, the authors show that if \(X\) is a \(G\)-space, then \(\overline{U}_G\) is a uniformity compatible with the topology on \(X\) if and only if for any \(x\in X\) and any \(O\in\mathbb{A}\), there exists a \(y\in X\) such that \(x\in \text{int cl}(Oy)\). They conclude by posing four open problems, the last of which is to determine whether every compactification of a space \(X\) is a \(G\)-compactification for a suitable group \(G\) acting on \(X\).NEWLINENEWLINEFor the entire collection see [Zbl 0983.00046].