  We define the local-triviality dimension for actions of compact quantum
groups on unital C*-algebras. The resulting compact quantum principal bundle is
said to be locally trivial when this dimension is finite. For commutative
C*-algebras, this notion recovers the standard definition of local triviality
of compact principal bundles. We prove that actions with finite
local-triviality dimension are automatically free. Then we apply this new
notion to prove the noncommutative Borsuk-Ulam-type conjecture under the
assumption that a compact quantum group admits a non-trivial classical subgroup
whose induced action has finite local-triviality dimension. This is a
noncommutative extension of the Borsuk-Ulam-type theorem for locally trivial
principal bundles.
