continuous an open image of a meager set is meager

Question

I want to know if the following is true. Let $X$ and $Y$ be topological spaces and $f\colon X\to Y$ a continuous open surjection. Suppose that $X$ is meager, then $Y$ is meager.

Recall that a meager set is a countable union of nowhere dense sets, and a set is nowhere dense if the interior of its closure is empty.

Any help will be appreciated.

Answer 1

I don't think it is true,

Let $X= \mathbb{Q}$ which is clearly meager & $Y$ is singleton. and let $f:X\to Y$ be constant map, so has satisfied all the desired properties, but $Y$ is not meager, because any subset is open in $Y$ since it is singleton.