Metric spaces

Editorially prepared introduction for this redesigned journal. This is new explanatory content, separate from Pablo's original notes.

1. Metric spaces

A metric space is a set together with a precise notion of distance. It provides a common language for convergence and continuity that is not limited to the real line.

Definition. A metric on a set \(X\) is a function \(d:X\times X\to[0,\infty)\) such that for all \(x,y,z\in X\), \(d(x,y)=0\) if and only if \(x=y\), \(d(x,y)=d(y,x)\), and \(d(x,z)\le d(x,y)+d(y,z)\).

Example. On \(\mathbb R^n\), the Euclidean metric is \(d(x,y)=\sqrt{\sum_{i=1}^n(x_i-y_i)^2}\). The discrete metric, defined by \(d(x,y)=0\) for \(x=y\) and \(1\) otherwise, is another metric on any set.

2. Open sets

The open ball about \(x\) of radius \(r>0\) is \(B_r(x)=\{y\in X:d(x,y)

3. Sequences and completeness

A sequence \((x_n)\) converges to \(x\) when \(d(x_n,x)\to0\). It is Cauchy if for every \(\varepsilon>0\), there exists \(N\) such that \(d(x_n,x_m)<\varepsilon\) for all \(n,m\ge N\).

Definition. A metric space is complete if every Cauchy sequence converges to a point of that space.

Contrast. \(\mathbb R\) with its usual distance is complete, while \(\mathbb Q\) is not: rational approximations to \(\sqrt 2\) form a Cauchy sequence without a rational limit.

4. Continuity

A map \(f:(X,d_X)\to(Y,d_Y)\) is continuous at \(x\) if for every \(\varepsilon>0\) there exists \(\delta>0\) such that \(d_X(x,y)<\delta\) implies \(d_Y(f(x),f(y))<\varepsilon\).

Metric spaces.

1. Metric spaces

2. Open sets

3. Sequences and completeness

4. Continuity