There seems to be something that I am fundamentally misunderstanding about quotient maps as I cannot convince myself that given a quotient map $f:X\to Y/R$ is open. To be specific, let $(X, \tau)$ be a topological space, $R$ an equivalence relation on $Y$ and $f:X\to Y/R$ a quotient map onto the equivalence classes where $(Y/R, \tau')$ has the quotient topology, i.e. $V\subset Y/R$ is open if and only if $f^{-1}\left[V\right]\subset X$ is open.
For $f$ to be an open map, we need to have that $U\subset X$ is open if and only if $f[U]\subset Y/R$ is open. But how do we know that any open subset of $X$ can be represented as a preimage of some subset of $Y/R$? By this I mean that why do we have the correspondence
$$U\subset X \text{ open} \Longleftrightarrow f^{-1}[V]\subset X \text{ open for some $V\subset Y/R$}$$
It could very well be that I have misunderstood how the definition of the quotient topology, but to me it seems that we only take those subsets of $Y/R$ whose preimage is open in $X$. But doesn't this leave the possibility that there is an open subset $U\subset X$ such that $f^{-1}[f[U]]$ is not open?
And conversely, how do we connect $f[U]\subset Y/R \text{ open}$ to $V\subset Y/R \text{ open}$, i.e. why is any open subset of $Y/R$ the image of some open subset of $X$?
