Defining a non-Hausdorff space

Question

I´d like to get a Hausdorff space such that some quotient be non-Hausdorff space, but I´ve doubts with this construction.

We take the real numbers $\mathbb{R}$ with the usual topology. We define the relation $x \sim y$ if and only if $x,y \in \mathbb{I}$ or $x,y \in \mathbb{Q}$, being $\mathbb{I}$ the irrational numbers and $\mathbb{Q}$ the rational numbers.

Now we get $X/\sim = \{[i],[q]\}$ where $[i]$ is the class of the irrational numbers, $[q]$ the class of the rational numbers.

Now, Have we obtained an example? There is, any open set of $X/\sim$ contains always $[i],[q]$?

Thank you all

Answer 1

Your example works. Any subset of $X/\sim$ is an open subset iff its pre-image in $\mathbb{R}$ (which can be either $\emptyset$, $\mathbb{I}$, $\mathbb{Q}$, or $\mathbb{R}$) is open, meaning the subset is open iff the pre-image (thus the subset) is either empty or full.

Answer 2

Two element trivial topological space.

The open sets of $\mathbb R/\sim$ are essentially the open subsets of $\mathbb R$ that are unions of equivalence classes. In your case there are only two equivalence classes $[i]$ and $[q]$ so there's only four possible unions we need to check. Namely the sets $\{\cdot\}$, $[q] = \mathbb Q$, $[i] = \mathbb R- \mathbb Q$ and $[i] \cup [q]= \mathbb R$.. So $\mathbb R/\sim$ has at most four different open subsets. The first and last are open (in $\mathbb R$) by definition of a topology. The middle two are not open (in $\mathbb R$) since they are both dense and disjoint. So $\mathbb R/\sim$ has exactly two different open sets.

As someone else mentioned your space is homeomorphic to the set $\{0,1\}$ with the topology $\tau = \{\varnothing ,\{0,1\}\}$. In general we can put the trivial topology on any set $X$ by saying the only open sets are $X$ and the empty set. It's called the trivial topology because it has the bare minimum number of sets required to be a topology and no more.

These spaces have no physical structure. You should think of giving $\mathbb R$ the trivial topology as collapsing all the points down to a singularity so every point is touching every other point.