# 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 pmWhy every singleton set is discrete and as well as indescrete?

Hira Rafiq

July 6@ 8:26 amLet 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.