Last update: 23 July 2014
A metric space is a set $X$ with a function $d:X\times X\to {\mathbb{R}}_{\ge 0}$ such that
(a)  if $x\in X$ then $d(x,x)=0,$ 
(b)  if $x,y\in X$ and $d(x,y)=0$ then $x=y,$ 
(c)  if $x,y\in X$ then $d(x,y)=d(y,x),$ 
(d)  if $x,y,z\in X$ then $d(x,y)\le d(x,z)+d(z,y)\text{.}$ 
Let $(X,d)$ be a metric space. Let $x\in X$ and $\epsilon \in $ {\mathbb{R}}_{>0}\text{.}$$ The ball of radius $\epsilon $ at $x$ is the set $${B}_{\epsilon}\left(x\right)=\{y\in X\hspace{0.17em}\hspace{0.17em}d(x,y)<\epsilon \}\text{.}$$
Let $(X,s)$ be a metric space.
The metric space uniformity on $X$ is the uniformity generated by the sets $${B}_{\epsilon}=\{(x,y)\in X\times X\hspace{0.17em}\hspace{0.17em}d(x,y)<\epsilon \}$$ for $\epsilon \in {\mathbb{R}}_{>0}\text{.}$
The metric space topology on $X$ is the topology generated by the sets $${B}_{\epsilon}\left(x\right)=\{y\in X\hspace{0.17em}\hspace{0.17em}d(x,y)<\epsilon \}$$ for $x\in X$ and $\epsilon \in {\mathbb{R}}_{>0}\text{.}$
Homework: Let $(X,d)$ be a metric space. Show that $X$ is Hausdorff.
Homework: Let $(X,d)$ and $(Y,\rho )$ be metric spaces and let $f:X\to Y$ be a function. Show that $f$ is uniformly continuous if and only if $f$ satisfies if $\epsilon \in {\mathbb{R}}_{>0}$ then there exists $\delta \in {\mathbb{R}}_{>0}$ such that if $x,y\in X$ and $d(x,y)<\delta $ then $\rho (f\left(x\right),f\left(y\right))<\epsilon \text{.}$
Homework: Let $(X,d)$ and $(Y,\rho )$ be metric spaces and let $f:X\to Y$ be a function. Show that $f$ is continuous if and only if $f$ satisfies if $\epsilon \in {\mathbb{R}}_{>0}$ and $x\in X$ then there exists $\delta \in {\mathbb{R}}_{>0}$ such that if $y\in X$ and $d(x,y)<\delta $ then $\rho (f\left(x\right),f\left(y\right))<\epsilon \text{.}$
Homework: Show that the function $f:\mathbb{R}\to \mathbb{R}$ given by $f\left(x\right)={x}^{2}$ is continuous but not uniformly continuous.
Let $(X,d)$ be a metric space and let $A\subseteq X\text{.}$
A totally bounded subset of $X$ is a subset $A\subseteq X$ such that if $\epsilon \in {\mathbb{R}}_{>0}$ then there exists $N\in {\mathbb{Z}}_{>0}$ and ${x}_{1},{x}_{2},\dots ,{x}_{N}\in X$ such that $$A\subseteq B({x}_{1},\epsilon )\cup B({x}_{2},\epsilon )\cup \cdots \cup B({x}_{N},\epsilon )\text{.}$$ A bounded subset of $X$ is a subset $A\subseteq X$ such that there exists $C\in {\mathbb{R}}_{>0}$ such that if $x,y\in A$ then $d(x,y)<C\text{.}$
Let $(X,d)$ be a metric space and let $A\subseteq X\text{.}$
(a)  If $A$ is compact then $A$ is totally bounded. 
(b)  If $A$ is totally bounded then $A$ is bounded. 
Proof.  
(a) Assume $A\subseteq X$ is compact.
(b) Assume $A\subseteq X$ is totally bounded.
$\square $ 
Homework: Let $X=\mathbb{R}$ with metric $d:X\times X\to {\mathbb{R}}_{>0}$ given by $$d(x,y)=\text{min}\{xy,1\}\text{.}$$ Show that $X$ is bounded but not totally bounded.
Homework: Let $A=(0,1)\subseteq \mathbb{R}$ where $\mathbb{R}$ has the standard metric $$d(x,y)=xy\text{.}$$ Show that $A$ is totally bounded but not compact.
(This is [BR, Theorem 2.37]) Let $X$ be a metric space and let $E$ be a subset of $X$. The set $E$ is compact if and only if every infinite subset of $E$ has a close point in $E$.
Proof. 

$\square $ 
Let $X$ be a metric space and let $E$ be a compact subset of $X$. Then $E$ is closed and bounded.
Proof. 

$\square $ 
These are a typed copy of handwritten notes from the pdf 140721UniformSpacesscanned140721.pdf.