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Coarse/Fine (Topology)**

Given a set $X$ and two topologies $\mathcal{T},{\mathcal{T}}^{\prime }$ on $X$, one says that $\mathcal{T}$ is coarser than ${\mathcal{T}}^{\prime }$ (or, equivalently, ${\mathcal{T}}^{\prime }$ is finer than $\mathcal{T}$) if $\mathcal{T}\subseteq {\mathcal{T}}^{\prime }$. 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 $\{\mathrm{\varnothing },X\}$ (the 'trivial' or 'indiscrete' topology), while the finest is $\mathcal{P}(X)$ (the 'discrete' topology).