Equicontinuity and Quasi-Uniformities

For topological spaces 𝑋, 𝑌 with a fixed compatible quasi-uniformity 𝑄 in 𝑌 and for a family (𝑓 𝑖 ) 𝑖∈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 𝑄-equicontinuity (i.e., equicontinuity with respect to the topology of 𝑋 and 𝑄) are compared. It is shown that 𝑄-equicontinuity implies even continuity, and if 𝑄 is locally symmetric, it implies topological equicontinuity too. It turns out that these notions are equivalent provided 𝑄 is a uniformity compatible with a compact topology, but the equivalence may fail even for a locally symmetric quasi-uniformity 𝑄 compatible with a compact metrizable topology.