Given a set \(X\), a topological basis on \(X\) is a collection \(\mathcal{B}\) of subsets of \(X\) satisfying:

  1. For any \(x \in X\), there is some \(B \in \mathcal{B}\) such that \(x \in B\).
  2. For any \(x \in X\) and \(B_1, B_2 \in \mathcal{B}\), if \(x \in B_1 \cap B_2\) then there is some \(B_3 \in \mathcal{B}\) with \(x \in B_3\) and \(B_3 \subseteq B_1 \cap B_2\).

We proved that given a topological basis \(\mathcal{B}\) on \(X\), the collection \(\tau\) of arbitrary unions of elements of \(\mathcal{B}\) is a topology on \(X\). We say that \(\mathcal{B}\) is a basis for \(\tau\), or generates \(\tau\).

We saw that the collections \(\{(a, b) \mid a < b \in \mathbb{R}\}\) and \(\{(p, q) \mid p < q \in \mathbb{Q}\}\) form bases for the standard topology on the real line, however \(\{(n, m) \mid n < m \in \mathbb{Z}\}\) does not.

Given a set \(X\), a metric on \(X\) is a function \(d : X \times X \rightarrow \mathbb{R}^{\ge 0}\) satifying:

  1. For any \(x, y \in X\), we have \(d(x, y) = 0\) if and only if \(x = y\).
  2. For any \(x, y \in X\), we have \(d(x, y) = d(y, x)\).
  3. (Triangle inequality) For any \(x, y, z \in X\), we have \(d(x, z) \le d(x, y) + d(y, z)\).

Given a point \(x \in X\) and an episilon \(\epsilon > 0\), we define the \(\epsilon\)-ball around \(x\) to be the set

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

We showed the collection \(\{B_d(x, \epsilon) \mid x \in X, \epsilon > 0\}\) forms a topological basis. We say that \(d\) generates this topology.

Recall the standard (or Euclidean) metric on \(\mathbb{R}\) is the metric \(d(x, y) = \|x - y\|\). This metric generates the standard topology.

Follow-up questions

Please spend a few minutes thinking about the following questions before the next lecture. If you are stuck, don’t worry, but don’t look the answer up.

  • While the collection \(\{(n, m) \mid n < m \in \mathbb{Z}\}\) does not form a basis for the standard topology on the real line, does it form a topological basis? If so what topology does it generate?
  • Given a set \(X\), can you think of a metric that generates the discrete topology on \(X\)?
  • Can you find a few more metrics on the real line that are different from the Euclidean metric but nonetheless generate the same topology as the Euclidean metric? This suggests that the topology generated by a metric carries “less information” than the metric.