Quotient $\mathbb{C}^n/\mathbb{Q}^n$

Question

how could I prove that the natural projection $\mathbb{C}^n\to \mathbb{C}^n/\mathbb{Q}^n$ (endowed with the quotient topology) is an open map?

In particular, for $n=1$, is $\mathbb{C}/\mathbb{Q}$ a $T_1$-space? Is it compact?

Thank you in advance

Answer 1

Let $\pi$ be the natural projection. If $A$ is an open subset of $\mathbb{C}^n$, then$$\pi^{-1}\bigl(\pi(A)\bigr)=\bigcup_{Q\in\mathbb{Q}^n}(A+Q),$$which is the union of open setes and therefore an open set. So, $\pi$ is an open map.

The space $\mathbb{C}/\mathbb{Q}$ is not $T_1$, because in it if $V$ ise a neighborhood of $[0]$ and if $W$ is a neighborhood of $\left[\sqrt2\right]$, then $V\cap W\neq\emptyset$.

And $\mathbb{C}/\mathbb{Q}$ is not compact, because the open cover $\left\{\pi\bigl(D(ix,1\bigr)\,\middle|\,x\in\mathbb R\right\}$ has no finite subcover.