... designed to accomplish a definite objective.We shall be interested in getting appropriate notions of neighbourhood,distinguishability of points, continuity and, later, differentiability. How is ... afitting decomposition obtained? A first possibility might be to consider Swith all its subsets. This conception, though acceptable in principle, is tooparticular: it leads to a quite disconnected ... looselyconveying ideas of continuity and limit. Through a succession of abstractions,the mathematicians have formalized the idea of open set while inserting it ina larger, more comprehensive context....