Motivation:

In this course we are interested in spaces where we can measure distances between points. For example, in the plane \(\mathbb{R}^2\) we can measure the distance between two points \((x_1,y_1)\) and \((x_2,y_2)\) using the Pythagorean theorem: \[d\left((x_1,y_1),(x_2,y_2)\right)=\sqrt{(x_1-x_2)^2+(y_1-y_2)^2}.\] This is a special case of a more general concept called a metric space. We are interested in this property because it allows us to define a notion of closeness between points which it will be key to provide insights from a database of points.

Metric spaces is a concept part of the analysis and geometry area of mathematics. In order to understand this concept we need to introduce some definitions and examples.

Definition 1.1. A metric space is a non-empty set \(X\) together with a function \(d:X\times X\to \mathbb{R}\) such that for all \(x,y,z\in X\) the following properties hold:

1. \(d(x,y)\geq 0\) and \(d(x,y)=0\) if and only if \(x=y\) (non-negativity and identity of indiscernibles).

2. \(d(x,y)=d(y,x)\) (symmetry).

3. \(d(x,z)\leq d(x,y)+d(y,z)\) (triangle inequality).

The function \(d\) is called a metric or distance function.

Previous properties seems to be very abstract, but they are very natural. For example, in the plane \(\mathbb{R}^2\) the function \(d\) defined above satisfies all the properties of a metric.

Example 1.2. Let \(X=\mathbb{R}^n\) and \(d:\mathbb{R}^n\times \mathbb{R}^n\to \mathbb{R}\) be the function defined by \[d\left((x_1,x_2,\cdots,x_n),(y_1,y_2,\cdots,y_n)\right)=\sqrt{\sum_{j=1}^n (x_j-y_j)^2}.\] Then \(d\) is a metric on \(\mathbb{R}^n\) called the Euclidean metric or \(l_2\)-metric.

When \(n=1,\) it coincides with the usual way to measure distances in the real line \(\mathbb{R},\) and with \(n=2\) it coincides with the Pythagorean way to measure distances in the plane \(\mathbb{R}^2.\)

The existence of metrics is no only restricted to \(\mathbb{R}^n.\) In fact, we can define metrics on more general spaces. For example, graphs are a very important concept in topological data analysis and we can define metrics on them.

Example 1.3. Consider a (finite) set of points \(V\) in \(\mathbb{R}^n\) and \(E\) a set of edges connecting points in \(V.\) Let \(X=\Gamma(V,E)\) be the set containing the vertices and edges, called a graph. We can define a metric on \(V\) by \[d(x,y)=\min\left\{\sum_{i=1}^n d(x_i,x_{i+1})\mid x_1=x, x_n=y, x_i\in V\right\}\] where the minimum is taken over all the paths that join two different points. This metric is called the graph metric on \(X.\)

And there are more abstract metric spaces, for example the metric space of subsets of a given metric space.

Example 1.4. Let \((X,d)\) be a metric space. We can define a metric on the set of all subsets of \(X\) by \[d(A,B)=\inf\{d(x,y)\mid x\in A, y\in B\}\] where \(A\) and \(B\) are subsets of \(X.\) This metric is called the Hausdorff metric on the set of subsets of \(X.\) The Hausdorff metric is a very important concept in topological data analysis because it allows us to compare subsets of a given metric space and determine its closeness.

A non-empty set \(X\) can admit more than one metric, for example the set \(\mathbb{R}^n\) admits the Euclidean metric, the Manhattan metric, the Chebyshev metric, etc. In principle, if we change the metric on a given set \(X,\) the metric space \((X,d)\) will change. However, there are metrics that are not so different from each other and in subsequent lectures we will see this.

Definition 1.5. Let \(X\) be a non-empty set and \(d_1\) and \(d_2\) two metrics on \(X.\) We say that \(d_1\) and \(d_2\) are equivalent if there exist positive constants \(c_1\) and \(c_2\) such that for all \(x,y\in X\) \[c_1d_1(x,y)\leq d_2(x,y)\leq c_2d_1(x,y).\]

The previous definition is very important because it allows us to compare metrics on a given set. For example, the Euclidean metric and the Manhattan metric on \(\mathbb{R}^n\) are equivalent.

Example 1.6. Let \(X=\mathbb{R}^2,\) the following functions are metrics on \(X:\)

1. \(d_1\left((x_1,y_1),(x_2,y_2)\right)=|x_1-x_2|+|y_1-y_2|,\) called the Manhattan metric.

2. \(d_\infty \left((x_1,y_1),(x_2,y_2)\right)=\max\{|x_1-x_2|,|y_1-y_2|\},\) called the Chebyshev metric.

3. \(d_p\left((x_1,y_1),(x_2,y_2)\right)=\sqrt[p]{(x_1-x_2)^p+(y_1-y_2)^p},\) called the \(l_p\)-metric.

4. \(d_{cos} \left((x_1,y_1),(x_2,y_2)\right)=1-\frac{x_1 x_2+y_1 y_2}{(x_1^2+x_2^2)(y_1^2+y_2^2)}\) called the Cosine metric.

5. \(d_{ind} \left((x_1,y_1),(x_2,y_2)\right)=\begin{cases} 0 & \text{if } (x_1,y_1)=(x_2,y_2) \\ 1 & \text{if } (x_1,y_1)\neq (x_2,y_2) \end{cases},\) called the indiscrete metric.

There is a generalization of the previous example for any \(n\in \mathbb{N}\) and \(\mathbb{R}^n.\) In fact, there are more metrics on \(\mathbb{R}^n\) than the ones listed above, but at least we can affirm that the first three metrics are equivalent.

How does affect the choice of metric in a given set? Next session, we will discuss the geometry behind metrics and how it affects the notion of closeness between points.