Let \(\tau_1\) and \(\tau_2\) be topologies on a set \(X\). The topology \(\tau_1\) is finer (also stronger or larger) than \(\tau_2\) if \(\tau_2\subseteq\tau_1\), and coarser (also weaker or smaller) than \(\tau_2\) if \(\tau_1\subseteq\tau_2\). The topologies \(\tau_1\) and \(\tau_2\) are comparable if \(\tau_1\subseteq\tau_2\) or \(\tau_2\subseteq\tau_1\).
Let \(X\) be a set and let \(\mathrm{Top}(X)\) be the set of topologies on \(X\), partially ordered by inclusion. Then \(\mathrm{Top}(X)\) is a complete lattice [@Mun00]:
the infimum of a nonempty family \((\tau_i)_{i\in I}\) is the intersection \(\bigcap_{i\in I}\tau_i\);
the supremum of \((\tau_i)_{i\in I}\) is the topology generated by \(\bigcup_{i\in I}\tau_i\), the intersection of all topologies containing every \(\tau_i\);
the greatest element is the discrete topology \(\mathcal P(X)\) and the least element is the indiscrete topology \(\ts{\emptyset,X}\).
The union of two topologies need not be a topology. On \(X=\ts{a,b,c}\), the union of \(\ts{\emptyset,\ts{a},X}\) and \(\ts{\emptyset,\ts{b},X}\) contains \(\ts{a}\) and \(\ts{b}\) but not \(\ts{a,b}\), so the supremum is in general strictly larger than the union.
Let \(f\colon X\to Y\) be a map of sets, let \(\tau_1\subseteq\tau_2\) be topologies on \(X\), and let \(\sigma_1\subseteq\sigma_2\) be topologies on \(Y\).
If \(f\colon(X,\tau_1)\to(Y,\sigma_2)\) is continuous, then \(f\colon(X,\tau_2)\to(Y,\sigma_1)\) is continuous.
If \(f\colon(X,\tau_2)\to(Y,\sigma_1)\) is an open map, then \(f\colon(X,\tau_1)\to(Y,\sigma_2)\) is an open map.
If a sequence \((x_k)\) converges to \(x\) in \((X,\tau_2)\), then it converges to \(x\) in \((X,\tau_1)\).
For \(n\geq1\), the Zariski topology \(\tau_{\zar}\) on \(\CC^n\), whose closed sets are the common zero sets of families of polynomials, is strictly coarser than the Euclidean topology \(\tau_{\an}\). Zero sets of polynomials are closed in \(\tau_{\an}\) because polynomials are continuous, so \(\tau_{\zar}\subseteq\tau_{\an}\). The open unit ball is in \(\tau_{\an}\) but not in \(\tau_{\zar}\): every nonempty Zariski-open subset of \(\CC^n\) contains the complement of the zero set of a nonzero polynomial, which is dense in \(\tau_{\an}\) and hence not contained in the unit ball.