Let \(X\) be a set and let \(\tau_1\) and \(\tau_2\) be two topologies on \(X\) with bases \(\mathcal{B}_1\) and \(\mathcal{B}_2\) respectively. We checked that if for any \(\tau_1\)-set \(U \subseteq X\) and \(x \in U\), if there is some \(V \in \mathcal{B}_2\) such that \(x \in V \subseteq U\) then \(\tau_2\) is finer than \(\tau_1\).

Let \((X, d)\) be a metric space. Let

\[\hat{d} : X \times X \rightarrow \mathbb{R}^{\ge 0}\]

be defined by

\[\hat{d}(x, y) = \min(d(x, y), 1).\]

We verified in lecture that this is indeed a metric, but we didn’t check all the details that it generates the same topology on \(X\) as \(d\).

First, we confirm that the topology generated by \(\hat{d}\) is finer than the topology generated by \(d\). Fix arbitrary \(y \in X\) and \(\epsilon > 0\) and fix arbitrary \(x \in B_d(y, \epsilon)\). Choose \(\delta = \min(\epsilon - d(x, y), 1)\). We claim that \(B_{\hat{d}}(x, \delta) \subseteq B_d(y, \epsilon)\). Indeed, fix some arbitrary \(z \in B_{\hat{d}}(x, \delta)\). We have \(\hat{d}(x, z) < \delta\) and so \(\hat{d}(x, z) < 1\) which means \(\hat{d}(x, z) = d(x, z)\), thus in particular we have \(d(x, z) < \delta\). Therefore we have \(d(z, y) \le d(z, x) + d(x, y) < \delta + d(x, y) \le \epsilon - d(x, y) + d(x, y) = \epsilon\). This means \(z \in B_d(y, \epsilon)\) as desired.

On the other hand, fix arbitrary \(y \in X\) and \(\epsilon > 0\) and fix arbitrary \(x \in B_{\hat{d}}(y, \epsilon)\). Choose \(\delta = \epsilon - d(x, y)\). We claim that \(B_{d}(x, \delta) \subseteq B_{\hat{d}}(y, \epsilon)\). As before, fix some arbitrary \(z \in B_{d}(x, \delta)\) which means \(d(x, z) < \delta\). We have \(\hat{d}(z, y) \le \hat{d}(z, x) + \hat{d}(x, y) \le d(z, x) + d(x, y) < \delta + d(x, y) = \epsilon - d(x, y) + d(x, y) = \epsilon\). This means \(z \in B_{\hat{d}}(y, \epsilon)\) as desired.

You’ll prove a result on the second written assignment that gives a nicer way of proving that two metrics generate the same topology.

We also defined three metrics on \(\mathbb{R}\):

  1. The standard Euclidean metric: \(d((x_1, y_1), (x_2, y_2)) = \sqrt{(x_1 - y_1)^2 + (x_2 - y_2)^2}\)
  2. The Manhattan metric: \(d((x_1, y_1), (x_2, y_2)) = \|x_1 - y_1\| + \|x_2 - y_2\|\)
  3. The square distance: \(d((x_1, y_1), (x_2, y_2)) = \max(\|x_1 - y_1\|, \|x_2 - y_2\|)\)

You will check on the second written assignment that these are all indeed metrics and generate the same topology on \(\mathbb{R}^2\).