Why would one want to attempt to extend notions such as convergence and continuity to a setting even more abstract than metric spaces?
The answer is that, already at a very elementary level, one encounters phenomena that do not fit into the framework of metric spaces: pointwise convergence, for instance—the most basic notion of convergence there is for functions—cannot be described as convergence with respect to a metric (as we show in this chapter).
Convergence and continuity in the metric setting were based on a notion of “closeness” for points: two points were sufficiently close if their distance, measured through the given metric, was sufficiently small. Going beyond metric spaces and still being able to meaningfully speak of convergence and continuity therefore ought to be based on an axiomatized notion of closeness. Such an axiomatization exists (and is surprisingly simple): it lies at the heart of the concept of a topological space. (For technical reasons, we pursue a slightly different, but equivalent route.)
KeywordsTopological Space Prime Ideal Commutative Ring Open Cover Separation Property
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