I am learning topology and would like to know the difference between Homeomorphism and Quotient maps. I will be grateful if anyone can help me in any way.
2026-04-01 15:11:39.1775056299
Difference between homeomorphism and quotient map
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Quotient maps could be not injective. However, when you have a quotient map $f:X\to Y$ then $f$ induces a homeomorphism between the quotient space $X/\sim_f$ and $Y$, where $x_1\sim_fx_2$ iff $f(x_1)=f(x_2)$. Note that this relation is trivial when $f$ is injective (therefore a homeomorphism) and in this case clearly $X/\sim_f\cong X$.