notes / Studies in Real and Complex Analysis / chapter 1: Metric and Topological Spaces / From my notebook

N5 Recap: Metric Spaces, Continuity and Limits

Recall a metric space is a set with a distance function such that

Definition N5.1 (Open ball).

is open if it has an open ball around every point:

For metric spaces and , is continuous

Definition N5.2 (Limit of a sequence). in if for all there exists such that for all , .

every open subset contains for all large enough .