Metric space

We equip a set with metric which satisfies:

• Then is called a metric space.

We have the usual definitions of convergence of sequences and their usual properties …

On for and we define the open ball and closed ball

A set is open iff every point in has an open ball contained in . This is the metric induced topology. A set is closed iff is open.

Let closed. Suppose is a convergent sequence in s.t. and . Suppose . Then any open ball around will contain some point in . But then no open ball around is in , so is not open, so is not closed - contradiction. Similarly, if all convergent sequences in that are in are also convergent in , then is closed.

Complete

A subset is bounded if there are some and s.t. . Totally bounded

Some properties

• ConvergentCauchyBounded (but none of the other directions hold)
• If is a complete subspace of then is closed in
• If is a closed subspace of a complete space then is also complete.

Continuous function spaces Contraction mapping Picard-Lindelof Theorem Compact - Sequentially Compact - Totally Bounded