## Tomasz Kubiak, Mar\'{\i }a Angeles de Prada Vicente

*On quasi-uniform space valued semi-continuous functions *

Comment.Math.Univ.Carolin. 50,1 (2009) 125-133. **Abstract:**F. van Gool [Comment. Math. Univ. Carolin. {33} (1992), 505--523] has introduced the concept of lower semicontinuity for functions with values in a quasi-uniform space $(R,\Cal U)$. This note provides a purely topological view at the basic ideas of van Gool. The lower semicontinuity of van Gool appears to be just the continuity with respect to the topology $T(\Cal U)$ generated by the quasi-uniformity $\Cal U$, so that many of his preparatory results become consequences of standard topological facts. In particular, when the order induced by $\Cal U$ makes $R$ into a continuous lattice, then $T(\Cal U)$ agrees with the Scott topology $\sigma (R)$ on $R$ and, thus, the lower semicontinuity reduces to a well known concept.

**Keywords:** lower semi-continuity, quasi-uniformity, continuous lattice

**AMS Subject Classification:** 54C08, 54E15, 06B35, 54F05