A metric space is a set equipped with a metric, which is a function that defines a distance between any two elements of the set. Formally, a metric space is defined as a pair $(X, d)$, where:

• $X$ is a set.

• $d : X \times X \to \mathbb{R}$ is a metric on $X$, meaning a function that satisfies the following properties for all $x, y, z \in X$:

  1. Non-negativity: $d(x, y) \geq 0$ (The distance between any two points is non-negative).
  2. Identity of indiscernibles: $d(x, y) = 0$ if and only if $x = y$ (The distance between two points is zero if and only if they are the same point).
  3. Symmetry: $d(x, y) = d(y, x)$ (The distance from $x$ to $y$ is the same as from $y$ to $x$).
  4. Triangle inequality: $d(x, z) \leq d(x, y) + d(y, z)$ (The distance from $x$ to $z$ is less than or equal to the sum of the distances from $x$ to $y$ and from $y$ to $z$).

Example of a Metric Space

Consider the set $X = \mathbb{R}^2$, which is the set of all pairs of real numbers (i.e., the Euclidean plane). The standard Euclidean distance $d$ between two points $(x_1, y_1)$ and $(x_2, y_2)$ in $\mathbb{R}^2$ is defined by:

$d((x_1, y_1), (x_2, y_2)) = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2}$

This function $d$ satisfies all the properties of a metric, making $(\mathbb{R}^2, d)$ a metric space.

Do you want more details or have any questions?

Here are 5 related questions:

1. Can you explain the concept of open and closed sets in a metric space?
2. How does the concept of convergence work in a metric space?
3. What is a complete metric space, and can you give an example?
4. How does the concept of a norm relate to a metric space?
5. What are some non-Euclidean examples of metric spaces?

Tip: Understanding the properties of a metric is crucial for exploring more advanced concepts in analysis, such as continuity, compactness, and connectedness in metric spaces.