Give 4–6 different topologies on \(\mathbb R\).
Solution
Consider the following topologies on \(\mathbb{R}\):
- \(\mathcal{T}_1 = \{\emptyset, \mathbb{R}\}\), the indiscrete topology;
- \(\mathcal{T}_2\), the Euclidean topology, generated by the basis \(\{(a, b) : a < b\}\);
- \(\mathcal{T}_3\), the lower limit (Sorgenfrey) topology, generated by the basis \(\{[a, b) : a < b\}\);
- \(\mathcal{T}_4 = \{U \subseteq \mathbb{R} : \mathbb{R} \setminus U \text{ is finite}\} \cup \{\emptyset\}\), the cofinite topology;
- \(\mathcal{T}_5 = \{U \subseteq \mathbb{R} : \mathbb{R} \setminus U \text{ is countable}\} \cup \{\emptyset\}\), the cocountable topology;
- \(\mathcal{T}_6 = \mathcal{P}(\mathbb{R})\), the discrete topology.
1. \(\mathcal{T}_1\) and \(\mathcal{T}_6\) differ from each other and from \(\mathcal{T}_2,\ldots,\mathcal{T}_5\).
\(\mathbb{R} \setminus \{0\}\) lies in \(\mathcal{T}_2, \ldots, \mathcal{T}_6\) but not in \(\mathcal{T}_1\). The singleton \(\{0\}\) lies in \(\mathcal{T}_6\) but in none of \(\mathcal{T}_1, \ldots, \mathcal{T}_5\): its complement is uncountable, so \(\{0\} \notin \mathcal{T}_4, \mathcal{T}_5\), and it contains no interval \((a, b)\) or \([a, b)\) with \(a < b\), so \(\{0\} \notin \mathcal{T}_2, \mathcal{T}_3\).
2. \(\mathcal{T}_2 \neq \mathcal{T}_3\).
\([0, 1) \in \mathcal{T}_3\), but \([0, 1) \notin \mathcal{T}_2\) because no interval \((-\varepsilon, \varepsilon)\) with \(\varepsilon > 0\) lies in \([0, 1)\).
3. \(\mathcal{T}_2\) and \(\mathcal{T}_3\) differ from \(\mathcal{T}_4\) and \(\mathcal{T}_5\).
\((0, 1)\) lies in \(\mathcal{T}_2\) and \(\mathcal{T}_3\) (it is the union of the intervals \([1/n, 1)\)), but its complement is uncountable, so it lies in neither \(\mathcal{T}_4\) nor \(\mathcal{T}_5\).
4. \(\mathcal{T}_4 \subsetneq \mathcal{T}_5\).
Every finite set is countable, so \(\mathcal{T}_4 \subseteq \mathcal{T}_5\). The set \(\mathbb{R} \setminus \mathbb{N}\) has countably infinite complement, so it lies in \(\mathcal{T}_5\) but not in \(\mathcal{T}_4\).
5. Q.E.D.
Steps <1>1–<1>4 separate every pair among \(\mathcal{T}_1, \ldots, \mathcal{T}_6\), so these are six distinct topologies on \(\mathbb{R}\).