Metric Spaces

A metric space is a set $X$ equipped with a binary distance metric $d : X^2 \rightarrow \R$ satisfying the following properties:

$$ \textbf{Positive Definiteness. } d(x, y) \geq 0 \text{ with }d(x,y)=0 \iff x=y \\

\textbf{Symmetry. } d(x,y) = d(y,x) \hspace{50pt} \\

\textbf{Triangle Inequality. } d(x,y) \leq d(x,z) + d(z,y) \hspace{53pt} $$

Definition Gauntlet

For each of the following definitions, consider a point $p \in X$ and a subset $E \sube X$.

Terminology	Definition	Description
Open Ball	An open ball of $p$ is a set
$B_r(p)=\{q \in X \space	\space d(p,q) < r\}$
defined by a radius $r > 0$.	Image
A ball with a dotted boundary centered at a point.
Limit Point	A point $p$ is a limit point of $E$ if every open ball of $p$ contains a point $q \in E$ not equal to $p$.	Image
Also points inside the blob, but most often referred to as the points along the boundary of the blob.
Isolated Point	A point $p$ is an isolated point of a $E$ if $p \in E$ and $p$ is not a limit point.	Image
A point in a separated blob which is not near any other points.
Interior Point	A point $p$ is an interior point of a $E$ if there exists $B_r(p) \sub E$.	Image
A point which is not on the boundary of a blob.
Closed Set	A subset $E$ is closed if it contains all of its limit points.	Image
A blob with a solid boundary to indicate that limit points on the boundary are included.
Open Set	A subset $E$ is open if every point in $E$ is an interior point.	Image
A blob with a dotted boundary to indicate that limit points on the boundary are excluded.
Compliment	The compliment of $E$ is the set
$E^c = \{p \in X \space	\space p \notin E\}$.	Image
All the stuff not in the blob.
Bounded Set	A subset $E$ is bounded if there exists a real number $M > 0$ and point $p$ for which $E \sub B_M(p)$.	Image
There is a big enough circle to fit the blob inside.
Dense Set	A subset $E$ is dense in $X$ if every point in $X$ is a limit point of $E$ or belongs to $E$.	The blob is suffused into everything around it, like the marbling on a steak.
Perfect Set	A subset $E$ is perfect if $E$ is closed and every point in $E$ is a limit point of $E$.	The blob has a boundary and every point it contains is near another point.
Convex Set	A subset $E$ of a vector space $X$ is convex if for any points $x, y \in E$ and $\lambda \in [0,1]$, $\lambda x + (1-\lambda)y \in E$.	Image
The straight line segment between any two points in $E$ is entirely contained in $E$.
Neighborhood	A subset $E$ is a neighborhood of a point $p$ if it is an open set containing $p$.	A generic shape with a dotted boundary containing a point.

Miscellaneous Theorems

Every Open Ball is Open. Any open ball $B_r (p)$ in a metric space $X$ is an open subset of $X$.

</aside>

Limit Points are Surrounded by Infinitely Many Points. Any neighborhood of a limit point of a set $E \subseteq X$ contains infinitely many points of $E$.

Proof </aside>

Finite Sets have no Limit Points. If $E \sube X$ is finite, it has no limit points.

</aside>

Open-Closed Complement Equivalence. For a subset $E \sube X$, the following hold and are equivalent:

$E$ is open $\iff$ $E^c$ is closed.
$E$ is closed $\iff$ $E^c$ is open.

Proof </aside>

Openness-Closedness Under Intersection and Union. The following hold for a metric space $X$, where $E_\alpha$ are understood to be subsets of $X$:

$\{E _\alpha\}$ is a collection of open subsets $\implies$ $\bigcup {\alpha} E{\alpha}$ is open.
1. $\{E \} _{i=1}^n$ is a finite collection of open subsets $\implies$ $\bigcap {i=1}^n E{i}$ is open.
$\{E _\alpha\}$ is a collection of closed subsets $\implies$ $\bigcap {\alpha} E{\alpha}$ is closed.
1. $\{E \} _{i=1}^n$ is a finite collection of closed subsets $\implies$ $\bigcup {i=1}^n E{i}$ is closed. </aside>