④ Metric spaces

Let \(M\) be a set and let \(d\colon M \times M \to \mathbb{R}\) be a function. If, for all \(x,\,y,\,z\in M\), the following conditions hold:

1. \(d(x, y) \geq 0\), with \(d(x, y) = 0\) iff \(x = y\);

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

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

then \(d\) is called a metric on \(M\), and the pair \((M, d)\) is called a metric space.

Let \((M, d)\) be a metric space. Let \((a_n)\) be a sequence in \(M\), and let \(a \in M\). We say that \((a_n)\) _converges to _\(a\) if, for all \(\varepsilon > 0\), there exists \(N \in \mathbb{N}\) such that \begin{equation} d(a_m, a) \leq \varepsilon \quad \text{for all }n \in \mathbb{N} \text{ with }n \geq N.\notag \end{equation} Such a sequence is called a convergent sequence.

Let \((M, d)\) be a metric space. A sequence \((a_n)\) in \(M\) is Cauchy if for all \(\varepsilon > 0\), there exists \(N \in \mathbb{N}\) such that \begin{equation} d(a_m, a_n) \leq \varepsilon \quad \text{for all }m,\,n \in \mathbb{N} \text{ with }m, n \geq N.\notag \end{equation}

A metric space \((M, d)\) is complete if every Cauchy sequence in \(M\) converges in \(M\).

Let \((M, d)\) be a metric space. A function \(f\colon M \to M\) is called a contraction mapping if there exists \(\rho \in (0, 1)\) such that \begin{equation} d(f(x), f(y)) \leq \rho d(x, y) \quad \text{for all } x,\,y\in M.\notag \end{equation}

Let \((M, d)\) be a metric space. A subset \(S \subseteq M\) is compact if every sequence \((x_{m})\) in \(S\) has a subsequence that converges in \(S\).