\(\frac{1}{2}\)-homogeneity of \(n\)th suspensions (Q386177)

Let \(\mathcal{H}(X)\) denote the group of homeomorphisms of a space \(X\). An orbit of \(X\) is the action of \(\mathcal{H}(X)\) at a point of \(X\), namely \(\{ h(x): h\in \mathcal{H}(X)\}\). A space is said to be \(\frac{1}{n}\)-\(homogeneous\) if \(X\) has exactly \(n\) orbits, for a positive integer \(n\). For \(n=2\), certain conditions on a continuum \(X\) are known under which the suspension over \(X\) is \(\frac{1}{2}\)-homogeneous.NEWLINENEWLINEIn this paper, the authors give a positive answer to the following question whether the second suspension of a homogeneous, non-locally connected, compact space \(X\) is \(\frac{1}{2}\)-homogeneous. Then the authors also show that the same is true for the \(n\)th suspension of such a space.NEWLINENEWLINEThe authors show that the suspension of a space is first countable at its vertices if and only if the space is countably compact. Then they introduce some examples of countably compact spaces by the light of these observations. Moreover, the authors introduce a way to construct a special kind of homeomorphisms in suspensions, and use these homeomorphisms to prove their main results. The authors present conditions on a space \(X\) under which the suspension of a Hausdorff space is \(\frac{1}{2}\)- homogeneous. Then the authors pose five interesting question about the subject.