J. Munkres. Topology (2013) Chapter 2: Topological Spaces and Continuous Functions. 20: The Metric Topology

Definition

A metric on a set \(X\) is a function \(d : X^2 \rightarrow \mathbb{R} \) such that, for all \(x,y,z \in X\):

  1. \(d(x,y) \geq 0\)
  2. \(d(x,y) = 0\) iff \(x = y\)
  3. \(d(x,y) = d(y,x)\)
  4. \em{Triangle Inequality} \(d(x,z) \leq d(x,y) + d(y,z)\)

We call \(d(x,y)\) the distance between \(x\) and \(y\).

A metric space is a pair \((X,d)\) where \(X\) is a set and \(d\) is a metric on \(X\). We usually write \(X\) for the metric space \((X,d)\).

Definition

Let \(X\) be a metric space. Let \(x \in X\) and \(\epsilon > 0\). The open ball with centre \(x\) and radius \(\epsilon\) is

\[ B_d(x,\epsilon) = \{y \in X \mid d(x,y) < \epsilon \} \]

Definition

Let \(X\) be a metric space. The metric topology on \(X\) is the topology generated by the basis consisting of all the open balls.

Proposition

Let \(X\) be a metric space and \(U \subseteq X\). Then \(U\) is open if and only if, for all \(x \in U\), there exists \(\epsilon > 0\) such that \(B(x,\epsilon) \subseteq U\).

Example 1

For any set \(X\), the discrete metric on \(X\) is defined by \(d(x,y) = 0\) if \(x = y\), \(d(x,y) = 1\) if \(x \neq y\). The topology induced by this metric is the discrete topology.

Example 2

The standard metric on \(\mathbb{R}\) is defined by \(d(x,y) = |x-y|\). The topology induced by this metric is the standard topology on \(\mathbb{R}\).

Definition

A topological space \((X, \mathcal{T})\) is metrizable iff there exists a metric \(d\{ on \(X\) such that the metric topology induced by \(d\) is \(\mathcal{T}\).

Definition

Let \(X\) be a metric space. Let \(A \subseteq X\). Then \(A\) is bounded iff there exists \(M\) such that \(\forall x,y \in A. d(x,y) \leq M\). In this case, the diameter of \(A\) is defined to be

\[ \operatorname{diam} A := \sup \{ d(x,y) \mid x,y \in A \} \enspace . \]

Definition

Let \(d\) be a metric on \(X\). The standard bounded metric corresponding to \(d\) is the metric \(\overline{d}(x,y) = \min(d(x,y),1)\).

Theorem 20.1

Let \(d\) be a metric on \(X\). Then the standard bounded metric corresponding to \(d\) is a metric on \(X\) that induces the same topology as \(d\).

Corollary

Being bounded is not a topological property.

Definition

Let \(n\) be a positive integer. For \(\vec{x} \in \mathbb{R}^n\), define the norm of \(x\), \(\|x\|\), by

\[ \| (x_1, \ldots, x_n) \| := (x_1^2 + \cdots + x_n^2)^{1/2} \enspace . \]

The Euclidean metric on \(\mathbb{R}^n\) is defined by

\[ d(\vec{x},\vec{y}) := \| \vec{x} - \vec{y} \| \]

The square metric is defined by

\[ \rho(\vec{x}, \vec{y}) := \max(|x_1 - y_1|, \ldots, |x_n - y_n|) \enspace . \]

Lemma 20.2

Let \(d\) and \(d') be two metrics on the same set \(X\). Let \(\mathcal{T}\) and \(\mathcal{T}'\) be the topologies that they induce. Then \(\mathcal{T} \subseteq \mathcal{T}'\) if and only if, for all \(x \in X\) and \(\epsilon > 0\), there exists \(\delta > 0\) such that \(B_{d'}(x, \delta) \subseteq B_d(x, \epsilon)\).

Theorem 20.3

The Euclidean metric and the square metric both induce the same topology on \(\mathbb{R}^n\), which is the same as the product topology.

Definition

Let \(J\) be a set. The uniform metric \(\overline{\rho}\) on \(\mathbb{R}^J\) is defined by

\[ \overline{\rho}(\{x_j\}_{j \in J}, \{y_j\}_{j \in J}) = \sup_{j \in J} \overline{d}(x_j, y_j) \]

where \(\overline{d}\) is the standard bounded metric on \(\mathbb{R}\). The topology induced by the uniform metric is called the uniform topology.

Theorem 20.4

The uniform topology on \(\mathbb{R}^J\) is finer than the product topology and coarser than the box topology. These inclusions are strict iff \(J\) is infinite.

Theorem 20.5

Let \(\overline{d}\) be the standard bounded metric on \(\mathbb{R}\). Define the metric \(D\) on \(\mathbb{R}^\omega\) by

\[ D((x_n),(y_n)) = \sup_{i \in \mathbb{Z}_+} \frac{\overline{d}(x_i,y_i)}{i} \]

Then \(D\) induces the product topology on \(\mathbb{R}^\omega\).

From the exercises

Proposition

Let \(X\) be a metric space. Then \(d : X^2 \rightarrow \mathbb{R}\) is continuous. In fact, the metric topology on \(X\) is the coarsest topology such that \(d\) is continuous.

Definition

Let \(X\) be the set of all sequences \((x_n) \in \mathbb{R}^\omega\) such that \(\sum_n x_n^2\) converges. The \(l^2\)-topology on \(X\) is the topology induced by the metric

\[ d((x_n),(y_n)) = \left( \sum_n (x_n - y_n)^2 \right)^{1/2} \]

Proposition

The \(l^2\)-topology is strictly finer than the uniform topology and strictly coarser than the box topology.

Definition

The Hilbert cube is \([0,1] \times [0,1/2] \times [0,1/3] \times \cdots \subseteq \mathbb{R}^\omega\).

Proposition

The \(l^2\), uniform and product topologies agree on the Hilbert cube. The box topology is strictly finer.

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