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Keywords:
lower semi-continuity; quasi-uniformity; continuous lattice
lower semi-continuity; quasi-uniformity; continuous lattice
Summary:
F. van Gool [Comment. Math. Univ. Carolin. {\bf 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.
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