Local Basis at a Point and First Countable Spaces

Описание к видео Local Basis at a Point and First Countable Spaces

Given a point x in a topological space (X, T), a local basis for x is a collection B_x of neighborhoods of x such that any neighborhood of x contains a set from the collection B_x. Recall a set is countable if it’s in injection with the natural numbers. Say a topological space is first countable if you can find a countable local basis for each point of x (meaning B_x is a countable set). We show that a metric space is first countable, while the real line with the finite complement topology is not first countable.

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