I am studying quotient topology.Different books define the concept differently but I think I have got the crux of the concept.I want to verify whether I have understood properly and whether it is sufficient for me.
Let $X$ and $Y$ be two topological spaces and $f:X\to Y$ be a surjection.Then $f$ is called a quotient map if $U\subset Y$ is open iff $f^{-1}(U)\subset X$ is open.Now suppose $X$ is a topological space and $Y$ is a set such that $f:X\to Y$ is a surjection.Then the topology on $Y$ that makes the map $f$ into a quotient map is the topology $\mathcal T(f)=\{U\subset Y:f^{-1}(U) $ is open in $X\}$.It is known as the quotient topology.Now if we have an equivalence relation $\sim$ on $X$ and $X^*=X/\sim$ be the quotient set,then together with the quotient topology $X^*$ is called the quotient space.Now $\mathcal T(f)$ on $X^*$ is the largest topology that makes the map $f$ continuous because if $\tau$ is another topology on $X^*$ such that $f$ is continuous then $\tau\subset \mathcal T(f)$.
Is this understanding fine and enough?
Best Answer
Let $X$ be a topological space, $Y$ be a set and $f:X\to Y$ be a map of sets. We define the final topology $\mathscr{T}_f$ on $Y$ relative to $f$ to be the largest topology on $Y$ making $f$ continuous; this topology is exactely defined by $$\mathscr{T}_f=\{U\subseteq Y : f^{-1}(U) \ is \ open \ in \ X \}$$ Now consider an equivalence relation $\sim $ in $X$ and $\pi:X\to X/\sim$ the quotient map. The quotient topology in $X/\sim$ is defined to be the final topology on $X/\sim$ relative to $\pi$. Now if $X$ and $Y$ are topological spaces and $f:X\to Y$ is a surjection and a quotient map i.e, $U\subseteq Y$ is open iff $f^{-1}(U)\subseteq X$ is open. Then we define in $X$ the relation $\sim$ by: $x\sim x^\prime \Leftrightarrow f(x)=f(x^\prime)$. It is easy to see that $\sim$ is an equivalence relation and that $X/\sim$ endowed with the quotient topology is homeomorphic to $Y$.