On quasi-uniformly continuous functions and Lebesgue spaces

We show that every continuous function from a Lebesgue quasi-uniform space to a small-set symmetric quasi-uniform space is quasi-uniformly continuous. We also introduce a new notion of Lebesgue quasi-uniformity which involves the two topologies generated by a quasi-uniformity and its conjugate, namely a pair Lebesgue quasi-uniformity, We show that every bicontinuous function from a pair Lebesgue quasiuniform space to a quasi-uniform space is quasi-uniformly continuous. Pair Lebesgue quasi-uniform spaces have several interesting properties. Thus, we observe that a topological space admits only pair Lebesgue quasi-uniformities if and only if it is hereditarily compact and quasi-sober and that a T-1 topological space admits a pair Lebesgue quasi-uniformity if and only if it is paracompact.