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.

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