A spatiality criterion and an example of a quasitopology which is not a topology. (English) Zbl 0695.54002

Kříž, Igor; Pultr, Aleš

Houston J. Math. 15, No. 2, 215-234 (1989).

Frames isomorphic to lattices of open sets of topological spaces are called spatial or topologies. The authors investigate a spatial criterion for frames (called quasi-topologies) satisfying the following distributive law \[ (D):\quad \bigvee_{j\in J}\bigwedge F_ j=\bigwedge \{\bigvee_{j\in J}\alpha (j):\quad \alpha \in \prod_{j}F_ j\}, \] where \(F_ j\) is finite, J is arbitrary and \(\alpha\) (j) is the j-th coordinate of the element \(\alpha\) of the cartesian product.
Theorem. A frame A satisfies (D) iff each \(\alpha\in A\) is a meet of \(\alpha\)-semiprime elements. For Hausdorff frames the condition (D) is equivalent with spatiality. The authors give an example of a quotient of a free lower semilattice which is a quasitopology and not a topology.

Reviewer: B.F.Smarda

MSC:

54A05	Topological spaces and generalizations (closure spaces, etc.)
06D99	Distributive lattices

Keywords:

spatial criterion; frames; quasi-topologies; Hausdorff frames; free lower semilattice