Vtyjkyuk

Let $(Y, tau_Y)$ be a pseudo metric space, and let $tau_Y$ be the associated topology. Let $sim$ be the equivalence relation defined by $$x sim y text if d_Y(x,y)=0,$$
for $x, y in Y$. Set $X = Y/sim$, and let $(X, d_X)$ be the quotient metric space. Consider the mapping $P: Y rightarrow X$ defined by $x mapsto [x]$. Let $tau_D$ be the direct image of topology $tau_Y$ on $X$ via the mapping $P$, and let $tau_X$ be the topology of the metric space $(X, d_X)$. Show that $tau_D = tau_X$.

I'm not sure how to start on this problem. I think I need to prove that open sets of $tau_D$ are contained in open sets of $tau_X$ and vice versa. Open balls generate $tau_X$ because it's a metric topology, but the problem is that I don't know what the open sets of $tau_D$ are. Any assistance is appreciated.

I think you can prove it by showing $tau_D$ and $tau_X$ contain each other.

• $tau_Dsubset tau_X$. This is because each open ball in $tau_Y$ is mapped onto an open ball in $tau_ X$.




• $tau_Xsubset tau_D$. This is because $tau_D$ is the quotient topology, and $Pcolon (Y,tau_Y)to(X,tau_X)$ is continuous, since $P^-1(B_X(x,varepsilon))=bigcup_P(y)=xB_Y(y,varepsilon)$ for each $xin X$ and $varepsilon> 0$, where $B_X(x,varepsilon)$ stands for the open ball in $X$ centered at $x$ with radius $varepsilon$.