Criteria for holomorphic completeness

It is proved that a complex space which is countable at infinity is holomorphically complete if and only if the homology groups with compact supports for coherent analytic sheaves are trivial in the nonzero dimensions and the topological vector space of zero-dimensional homology with compact support of the structure sheaf is separated (Hausdorff). This result is then applied to complex spaces which can be represented as a union of an increasing sequence of holomorphically complete open sets and to complex spaces which locally admit holomorphically complete mappings into holomorphically complete spaces.