Certain Observations on a \({{\,\mathrm{U_{fin}}\,}}\)-Type Selection Principle

A weaker variant of the selection principle \({{\,\mathrm{U_{fin}}\,}}({\mathcal {O}},\Omega ),\) ${\mathrm{U}}_{\mathrm{fin}}(\mathcal{O},\mathrm{\Omega }),$ namely \({{\,\mathrm{U_{fin}}\,}}({\mathcal {O}},\overline{\Omega }),\) ${\mathrm{U}}_{\mathrm{fin}}(\mathcal{O},\overline{\mathrm{\Omega }}),$ is investigated in this article. We present situations where \({{\,\mathrm{U_{fin}}\,}}({\mathcal {O}},\Omega )\) ${\mathrm{U}}_{\mathrm{fin}}(\mathcal{O},\mathrm{\Omega })$ behaves differently from \({{\,\mathrm{U_{fin}}\,}}({\mathcal {O}},\overline{\Omega }).\) ${\mathrm{U}}_{\mathrm{fin}}(\mathcal{O},\overline{\mathrm{\Omega }}).$ Few characterization results are obtained by considering mappings into the Baire space. Several results are presented concerning critical cardinalities. In particular, we perform investigations assuming near coherence of filters (NCF) and semifilter trichotomy. Besides, \({{\,\mathrm{U_{fin}}\,}}({\mathcal {O}},\overline{\Omega })\) ${\mathrm{U}}_{\mathrm{fin}}(\mathcal{O},\overline{\mathrm{\Omega }})$ is characterized using weakly groupable and related covers. We also exhibit certain game theoretic observations.