# Coarser and Finer Topology

If ${\tau _1}$ and ${\tau _2}$ are two topologies defined on the non empty set X such that ${\tau _1} \subseteq {\tau _2}$, i.e. each member of ${\tau _1}$is also in ${\tau _2}$, then ${\tau _1}$is said to be coarser or weaker than ${\tau _2}$ and ${\tau _2}$ is said to be finer or stronger than ${\tau _1}$.

It may be noted that indiscrete topology defined on the non empty set X is the weakest or coarser topology on that set X and discrete topology defined on the non empty set X is the stronger or finer topology on that set X.

Note: The topology which is both discrete and indiscrete such topology which has one element in set X. i.e. X = {a}, $\tau =${$\phi$, X}. Every singleton set is discrete as well as indiscrete topology on that set.