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Coarse/Fine (Topology)**

Given a set \(X\) and two topologies \(\mathcal{T},\mathcal{T}'\) on \(X\), one says that \(\mathcal{T}\) is coarser than \(\mathcal{T}'\) (or, equivalently, \(\mathcal{T}'\) is finer than \(\mathcal{T}\)) if \(\mathcal{T}\subseteq\mathcal{T}'\). This turns the set of all topologies on \(X\) into a partial order. One often seeks the coarsest topology which satisfies a given condition, for example the product topology is the coarsest topology on \(X\times Y\) which makes both \(\pi_1:X\times Y\to X,\:(x,y)\mapsto x\) and \(\pi_2:X\times Y\to Y,\:(x,y)\mapsto y\) continuous. The coarsest topology on any set \(X\) is \(\{\emptyset,X\}\) (the 'trivial' or 'indiscrete' topology), while the finest is \(\mathcal{P}(X)\) (the 'discrete' topology).