Malʹtsev and retral spaces

A space X is Malʹtsev if there exists a continuous map M : X3 → X such that M(x, y, y) = x = M(y, y, x). A space X is retral if it is a retract of a topological group. Every retral space is Malʹtsev. General methods for constructing Malʹtsev and retral spaces are given. An example of a Malʹtsev space which is not retral is presented. An example of a Lindelöf topological group with cellularity the continuum is presented. Constraints on the examples are examined.