Metric Spaces Review
We will now review some of the definitions of a metric and a metric space and review some examples of metric spaces that we saw recently.
On the Metric Spaces page we first defined a special type of function on a set $M$ known as a Metric which is a function $d : M \times M \to [0, \infty)$ which takes each pair $(x, y)$ and maps it to some nonnegative real number called the Distance between $x$ and $y$. We say such a function $d$ is a metric of $M$ if it satisfies the following three properties:
The first property is that for all $x, y \in M$ we must have that $d$ is symmetric:
\begin{align} \quad d(x, y) = d(y, x) \end{align}
The second property that $d$ must have is that the distance from $x$ to $y$ equals $0$ if and only if $x$ and $y$ are the same point, that is:
\begin{align} \quad d(x, y) = 0 \quad \Leftrightarrow x = y \end{align}
The third property is known as the triangle inequality. It says that if $z$ is any intermediary point, then the distance from $x$ to $y$ must be less than or equal to the distance from $x$ to $z$ plus the distance from $z$ to $y$. In other words, the distance of any non-direct "path" from $x$ to $y$ is always greater than or equal to the distance of the direct path from $x$ to $y$. So, for all $x, y, z \in M$ we must have that:
\begin{align} \quad d(x, y) \leq d(x, z) + d(z, y) \end{align}
If $d$ is a metric as summarized above, then the set $M$ with a metric $d$ defined on $M$ is called a Metric Space and is denoted as the pair $(M, d)$, or sometimes simply as $M$ for brevity, and if $S \subseteq M$ then $(S, d)$ is said to be a Metric Subspace of $(M, d)$ (with the metric $d$ restricted to elements in $S$).
On the Some Metrics Defined on Euclidean Space page we looked at an important metric defined for all $\mathbf{x} = (x_1, x_2, ..., x_n), \mathbf{y} = (y_1, y_2, ..., y_n) \in \mathbb{R}^n$ by:
\begin{align} \quad d(\mathbf{x}, \mathbf{y}) = \sum_{k=1}^{n} \mid x_k - y_k \mid \end{align}
On The Chebyshev Metric page we looked at another important metric defined for all $\mathbf{x}, \mathbf{y} \in \mathbb{R}^n$ known as the Chebyshev Metric given by:
\begin{align} \quad d(\mathbf{x}, \mathbf{y}) = d(\mathbf{x}, \mathbf{y}) = \max_{1 \leq k \leq n} \{ \mid x_k - y_k \mid \} \end{align}
On The Discrete Metric page we looked at a more abstract metric known as the Discrete Metric defined for all $x, y \in M$ ($M$ an arbitrary set) by:
\begin{align} \quad d(x, y) = \left\{\begin{matrix} 0 & \mathrm{if} \: x = y\\ 1 & \mathrm{if} \: x \neq y \end{matrix}\right. \end{align}
We looked at another abstract metric on the The Standard Bounded Metric known as the Standard Bounded Metric defined for all $x, y \in M$ and with respect to any other metric $d$ by:
\begin{align} \quad \bar{d}(x, y) = \mathrm{min} \{ 1, d(x, y) \} \end{align}
After looked at all of those examples we then looked at a generalization of the triangle inequality property of a metric on The Polygonal Inequality for Metric Spaces page known as the Polygonal Property which says that if $(M, d)$ is a metric space and $x_1, x_2, ..., x_m \in M$ then:
\begin{align} \quad d(x_1, x_m) \leq \sum_{k=1}^{m-1} d(x_k, x_{k+1}) \end{align}