Open mapping from normal space to T1 space

Question

There is an exercise in Engelking`s "General topology" (p. 49, ex. 1.5.M), to give an example of an open surjective mapping of a normal space onto a T1 space that is not a T2. Please, help me.

Answer 1

A rather trivial example:

$f: X=(\Bbb R, \mathcal{T}_i) \to Y=(\Bbb R, \mathcal{T}_{cf})$ defined by $f(x)=x$ and $\mathcal{T}_i$ the indiscrete (aka trivial) topology and $\mathcal{T}_{cf}$ the cofinite topology. $X$ is normal (trivially, as the only closed sets are $\Bbb R$ and $\emptyset$) and $f$ an open bijection (likewise trivial), while $Y$ is a well-known example of a $T_1$ but not $T_2$ space.

A less trivial, stronger and altogether more interesting example is to take $X= \Bbb R \times \{0,1\}$ as a subspace of the plane, and to take $f$ as the quotient map that identifies all pairs $\{(x,0), (x,1)\}$ where $ x \neq 0$, to a point and leaves the two origins alone. $Y$ is the resulting quotient space, which is $T_1$ but not $T_2$ (line with two origins) and the domain is metric so normal in the stronger sense. $f$ is easily seen to be both open and continuous.

Answer 2

The inclusion between $j:(\mathbb R, T_{CF})\twoheadrightarrow(\mathbb R, T_u)$. Beeing $T_{CF}$ the co-finite topology and $T_u$ the usual topology, both in $\mathbb R$.