On Wilker-Type Properties

Wilker-type properties record how a compact set can be split along a binary open cover, and they are precisely the continuity conditions for the union operation on open lattices, governing the topology of power and function spaces. We give a unified treatment of the four properties , , and . We prove that and are hereditary for arbitrary subspaces, whereas and are hereditary only for closed subspaces; all four are preserved by continuous retracts, and , , are finitely additive over topological sums. For KC-spaces and coincide, and spaces whose compact subspaces are Hausdorff belong to . Using Kovár's compact KC-space M₀ and its variant ⁺, we obtain countably infinite compact KC-spaces in , showing is strictly weaker than . Conversely, we construct compact T₁ spaces whose product fails , proving the classes and are not reflective. We further show Vietoris constructions need not preserve , and prove that (M₀) (M₀,) is not sober and fails property~S. Open problems are collected at the end.