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The basic ideas in elementary real analysis are *convergence* and *continuity*.

These can be phrased in a more general context than in terms of **R** and have proved useful in many areas of both pure and applied mathematics.

The most straightforward generalisation is to a *metric space* but even this proved to be insufficient to satisfy the demands of those who needed to apply these ideas and so the ultimate abstraction is to the concept of a *topological space*.

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