A necessary and sufficient condition for the complements of a general kind of topological sphere arrangement (and those of all its subarrangements) to have homologically trivial components is that all intersection degeneracies be nonnegative in a certain sense. Transverse intersections are not assumed. This extends the domain of application of formulas which count these components in terms of degeneracies. In dimension 2 we show that the union of pseudocircles in an arrangement with possibly nontransverse intersections is the same as the union of an arrangement with only transverse intersections.
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