  We study the time-reversal odd quark fragmentation function and its
consequences on the hard processes. T-odd quark fragmentation function may
arise, although QCD is T-invariant, from the non-perturbative dynamics in the
fragmentation process. Assuming the factorization, we extract the T-odd
fragmentation function from the inclusive pion production in the transversely
polarized proton-proton collision. We then estimate the single spin asymmetry
$A_{OT}$ of the semi-inclusive deep inelastic scattering with unpolarized
lepton and the transversely polarized proton. We also discuss the meson cloud
effects on the transversity distribution $h_1(x)$ of the nucleon. Asymmetry for
the $\pi^-$ production is found to be much smaller than the naive expectation,
since the $d$-quark transversity is considerably suppressed by the pion cloud
effects
