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.

Explore the concept of metric spaces with a focus on definitions, properties, and examples. Learn about the standard Euclidean distance formula and how it satisfies the properties of a metric. Ideal for advanced learners and those interested in mathematical analysis.

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