Quasiorders, principal topologies, and partially ordered partitions

Abstract

ABSTRACT. The quasiorders on a set X are equivalent to the topologies on X which are closed under arbitrary intersections. We consider the quaisorders on X to be partial orders on the blocks of a partition of X and use this approach to survey some fundamental results on the lattice of quasiorders on X.

Untitled

“…This correspondence 26 TYLER CLARK AND TOM RICHMOND dates back to [Alexandroff 1937]. (See [Richmond 1998] for a survey of this connection.) One approach to counting the convex topologies would be to find a (biordered) characterization of convex topologies using some compatibility between the specialization order and the given total order.…”

On some topological properties in the class of Alexandroff spaces

“…In fact, since the eighties of the 20th century, the interest in Alexandroff spaces was a consequence of the very expanding role of finite spaces in digital topology and its applications in biology, social and natural sciences, digital image processing etc. For some elementary properties of Alexandroff spaces we refer to [2,21].…”

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“…Since computer arithmetic and computer graphics are based on finite sets of machine numbers or pixels, computer applications have driven much interest in Alexandroff spaces. Elementary properties of Alexandroff spaces are given in [2,3,7]. From a different perspective, Uzcátegui and Vielma [9] studied Alexandroff spaces viewed as closed sets of the Cantor cube 2 X .…”