\part{Metric spaces}


We now return to metric spaces using our new fancy knowledge of topological spaces.


\begin{theorem}
A metric space $(X,d)$ is a topological space $(X,\mathcal{T})$ generated by its open balls as a basis.
\end{theorem}

We had designed topologies to be a generalization of metric spaces, so all metric spaces form a topological space. However, given a topology, when can one find a metric to generate said topology? This is the class of \emph{metrizable topological spaces}; topological spaces that can be reformulated as metric spaces.

\begin{proposition}
Let $(X,d)$ be a metric space. The open balls of $(X,d)$ are a basis for a topological space $(X,\mathcal{T})$. The resulting topological space is called the \emph{topological space induced by $(X,d)$}.
\end{proposition}

\begin{definition}
A \emph{metrizable topological space} is a topological space $(X,\mathcal{T})$ such that there exists a metric $d : X\times X \to [0,\infty)$ such thatthe topology induced by $d$ is $\mathcal{T}$.
\end{definition}

Different metrics on the same set can possibly induce the same topology; they are said to be equivalent metrics if they both form the same topology.
-equivalent metrics

The topologies induced by metric spaces share the property of being \emph{Hausdorff spaces}, that is, any two points have disjoint neighborhoods. Topologies can be studied by examining how 'separated' its points are; more on this later.

- topologies induced by metric space are hausdorff


\chapter{Limits in metric spaces}


Though convergence of sequences may be established without a metric space, metric spaces provide a more intuitive definition for convergence.


\begin{definition}
In a metric space $(X,d)$, a \emph{limit} of a sequence is a point $p$ that is arbitrarily close all remaining terms of a sequence. A \emph{convergent sequence} is a sequence with a limit.
\[ (X,d) \]
\[ \lim_{n\to \infty} x_n =p  \iff \forall \varepsilon \in (0,\infty) [ \exists N \in \mathbb{N} [ n > N \implies d(x_n , p) < \varepsilon  ]  ]\]
\[ \lim_{n\to \infty} x_n =p  \iff \forall \varepsilon \in (0,\infty) [ \exists N \in \mathbb{N} [ n > N \implies x_n \in B_{X}(p,\varepsilon )  ]  ]\]
\end{definition}

\begin{proposition}
\[\lim_{n \to \infty} x_n = x \land  \lim_{n \to \infty} x_n = y \implies x=y \]
\end{proposition}



Limits can be used to express continuous functions between metric spaces
\begin{proposition}
\end{proposition}


- same sequence converging to x and y means x=y


-cauchy sequence
-subsequence
-bolzano weierstrass therem topology

\chapter{Complete metric space}

$(X,d)$ is complete iff cauchy sequences converge in X


\chapter{Isometry}

Topological properties are conserved under homeomorphisms;




\chapter{Banach fixed-point theorem}
- Contraction mapping
- Banach fixed point theorem
- Allude to applications in numerical analysis
\chapter{Baire spaces}


\chapter{Metrizations}

Metric spaces are much nicer to work with than topological spaces, so given a topolgical space, mathematicians often want to see if it is metrizable.

We search for a set of minimal properties that a topological space can obey that can be used to construct a metric. Such results are called \emph{metrization theorems}; sufficient conditions for a topological space to be metrizable.

Note that this study makes use of the separation properties.



\begin{theorem}[Urysohn Metrization theorem]
\end{theorem}


