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