  It is very difficult to obtain a realistic model of a closed inflationary
universe. Even if one fine-tunes the total number of e-folds to be sufficiently
small, the resulting universe typically has large density perturbations on the
scale of the horizon. We describe a class of models where this problem can be
resolved. The models are unattractive and fine-tuned, so the flatness of the
universe remains a generic prediction of inflationary cosmology. Nevertheless
one should keep in mind that with the fine-tuning at the level of about one
percent one can obtain a semi-realistic model of a closed inflationary
universe. The spectrum of density perturbations in this model may have a
cut-off on the scale of the horizon. Similar approach may be valid in
application to a compact inflationary universe with a nontrivial topology.
