TITLE: Canonical projection to subspace is a homeomorphism?
QUESTION [1 upvotes]: Let $X$ be a topological space, $V$ a closed set, and $U$ and open set such that $V\subset U \subset X$. Consider the quotient topology and let $\pi$ be the canonical projection into $X/V$. 
I want to show that $pr=\pi|_{U-V}:U-V\rightarrow (U-V)/V$ is a homeomorphism. It feels like it might be a general thing that $\pi$ is a homeomoprhism. I am having trouble showing that $pr$ is a homemorphism. 
Attempt: Continuity is clear since $\pi$ is continuous. It is clearly surjective since we have restricted the domain to the image. To show injectivity, assume $pr$ is not injective. Then for some $x\neq y \in U-V$ we have $pr(x)=pr(y)$. But this implies $x$ and $y$ were in $V$ giving a contradiction. 
I am not sure how to show that the inverse is continuous. And I feel like there is an easier way to approach this.

REPLY [1 votes]: Not only is this not false, it is trivially true. 
$U-V$ is open in $X$ if and only if $\pi(U-V)$ is open in $X/V$
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