We prove that a semiregular topological space $X$ is completely regular if and only if its topology is generated by a normal quasi-uniformity. This characterization implies that each regular paratopological group is completely regular. This resolves an old problem in the theory of paratopological groups, which stood open for about 60 years. Also we define a natural uniformity on each paratopological group and using this uniformity prove that each (first countable) Hausdorff paratopological group is functionally Hausdorff (and submetrizable). This resolves another two known open problems in the theory of paratopological groups.

The topological structure of (homogeneous) spaces and groups with countable cs∗-character

We present characterizations of metric spaces that are micro-, macro-or bi-uniformly equivalent to the extended Cantor set EC D ˚P1 iD n 2x i 3 i j n 2 N; .x i / i2Z 2 f0; 1g Z « R, which is bi-uniformly equivalent to the Cantor bi-cube 2 <Z D f.x i / i2Z 2 f0; 1g Z j there exists n such that x i D 0 for all i ng endowed with the metric d..x i /; .y i // D max i2Z 2 i jx i y i j. The characterizations imply that any two (uncountable) proper isometrically homogeneous ultrametric spaces are coarsely (and bi-uniformly) equivalent. This implies that any two countable locally finite groups endowed with proper left-invariant metrics are coarsely equivalent. For the proof of these results we develop a technique of towers which may be of independent interest.