  In the seminal monograph "Theory of retracts", Borsuk raised the following
question: suppose two compact ANR's are $h$--equal, i.e. mutually homotopy
dominate each other, are they homotopy equivalent? The current paper approaches
this question in two ways. On one end, we provide conditions on the fundamental
group which guarantee a positive answer to the Borsuk question. On the other
end, we construct various examples of compact $h$--equal, not homotopy
equivalent continua, with distinct properties. The first class of these
examples has trivial all known algebraic invariants (such as homology, homotopy
groups etc.) The second class is given by $n$--connected continua, for any $n$,
which are infinite $CW$--complexes, and hence ANR's, on a complement of a
point.
