# Comparing Topologies

It's possible that a set $X$ can be endowed with two or more topologies that are comparable. Over the years, mathematicians have used various words to describe the comparison: a topology $\tau_1$ is said to be **coarser** than another topology $\tau_2$, and we write $\tau_1\subseteq\tau_2$, if every open set in $\tau_1$ is *also* an open set in $\tau_2$. In this scenario, we also say $\tau_2$ is **finer** than $\tau_1$. But other folks like to replace "coarser" by "smaller" and "finer" by "larger." Still others prefer to use "weaker" and "stronger." But how can we keep track of all of this? Personally, I like to think in terms of (and while sipping a cup of) coffee!

(Now does it make sense why the indiscrete topology on a set $X$ is the coarsest/smallest/weakest topology, while the discrete topology is the finest/largest/strongest topology? )