Certain observations on tightness and topological games in bornology (Q6609583)

In this article the authors investigate the function space \(C(X)\) with respect to the topology \(\tau ^\mathcal{S}_ \mathcal{B}\) of strong uniform convergence on a bornology \(B\) on a set \(X\). They mainly focus on some variations on the tightness property, \(k\)-spaces, the discretely selective property and related games in the function space \((C(X),\tau ^\mathcal{S}_\mathcal{B})\) on a bornology. It is shown that the tightness and the supertightness properties of \((C(X), \tau^\mathcal{S}_\mathcal{B})\) are interchangeable. The Id-fan tightness and the T-tightness of \((C(X), \tau^\mathcal{S}_\mathcal{B})\) are characterized in terms of bornological covering properties of \(X\). \N\NThen they prove that whenever \((C(X), \tau^\mathcal{S}_\mathcal{B})\) is a \(k\)-space it is equivalent to a selection principle related to bornological covers of \(X\). At the end the authors introduce the notions of strong \(\mathcal{B}\)-open game and \(\gamma_{\mathcal{B}^\mathcal{S}}\)-open game on \(X\) and obtain their consonances with other classical games on \(X\).\N\N Later they study the discretely selective property and associated games. Under a certain condition on \(\mathcal{B}\), \((C(X), \tau^\mathcal{S}_\mathcal{B})\) is shown to be discretely selective. Several interactions between the Gruenhage game on \((C(X), \tau^\mathcal{S}_\mathcal{B})\), topological games on \((C(X), \tau^\mathcal{S}_\mathcal{B})\) related to the discretely selective property and certain games on \(X\) are also presented.