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.
Ahmad Shah
November 6 @ 8:04 pm
Why every singleton set is discrete and as well as indescrete?
Hira Rafiq
July 6 @ 8:26 am
Let X={a}
P(X)= {∅,{a}}
τ={∅,{a}} →(1)
Indiscrete Topology=Topology made by ∅ & ground set itself
Discrete Topology=Topology made by all possible subsets
Clearly, (1) is both.