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It is proved that there exists a fixed point index theory foroperators which are condensing on the countable subsets of the space only.Even weaker compactness assumptions on countable subsets suffice,e.g. conditions with respect to classes of measures of noncompactness,or if measures of noncompactness of countable noncompact sets arenot preserved (not necessarily decreased).As an application, we prove a generalization of the Fredholm alternative.

Fixed point theorems and fixed point index for countably condensing maps