A topological inverse semigroup is a Hausdorff topological space together with a continuous multiplication and an inversion. With every topological inverse semigroup \(S\) one can associate its band of idempotents \(E(S)\). This paper studies compact topological inverse semigroups by relating them to their band of idempotents. Next, we describe briefly some of the main results of the paper. Theorem 4.1 proves (among several other things) that a topological inverse semigroup \(S\) has open right principal ideals if and only if its band \(E(S)\) is a semigroup with open principal ideals if and only if \(E(S)\) is a semigroup with pseudo-open translations. Semigroups satisfying this theorem are called bopi-semigroups. Theorem 4.6 proves that a first countable compact inverse bopi-semigroup is necessarily metrizable. This is obtained as a consequence of the results on cardinal invariants for compact commutative bands with open principal ideals obtained in Section 2. Another example of how the relation between compact topological inverse semigroups and their bands is exploited in this paper is Proposition 4.10. There, the author proves that the one-point Alexandroff compactification \(\mathcal{A}(X)\) of an uncountable discrete space \(X\) admits no structure of topological inverse bopi-semigroup. The proof relies basically upon the fact, proved in Section 2, that \(\mathcal{A}(X)\) admits no structure of topological commutative band with open principal ideals.