A better framework for first countable spaces
Keywords:First axiom of countability, Second axiom of countability, Countably compact, Sequentially compact, Sequentially complete, Continuous convergence, Sequentially continuous, Semiuniform convergence spaces, Convergence spaces, Filter spaces, Topological spac
In the realm of semiuniform convergence spaces first countability is divisible and leads to a well-behaved topological construct with natural function spaces and one-point extensions such that countable products of quotients are quotients. Every semiuniform convergence space (e.g. symmetric topological space, uniform space, filter space, etc.) has an underlying first countable space. Several applications of first countability in a broader context than the usual one of topological spaces are studied.
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