E. Corbacho, V. Tarieladze, R. Vidal

Equicontinuity and Quasi-Uniformities

Abstract:
For topological spaces $X,Y$ with a fixed compatible quasi-uniformity $\mathcal Q$ in $Y$ and for a family $(f_i)_{i\in I}$ of mappings from $X$ to $Y$, the notions of even continuity in the sense of Kelley, topological equicontinuity in the sense of Royden and $\mathcal Q$-equicontinuity (i.e., equicontinuity with respect to the topology of $X$ and $\mathcal Q$) are compared. It is shown that $\mathcal Q$-equicontinuity implies even continuity, and if $\mathcal Q$ is locally symmetric, it implies topological equicontinuity too. It turns out that these notions are equivalent provided $\mathcal Q$ is a uniformity compatible with a compact topology, but the equivalence may fail even for a locally symmetric quasi-uniformity $\mathcal Q$ compatible with a compact metrizable topology.

Keywords:
Topological space, quasi-uniform space, even continuity, topological equicontinuity, uniform equicontinuity.

MSC 2000: Primary: 54C35; secondary: 54E15