Complete regular non-normal Hausdorff space
The Moore plane, also sometimes called the Niemytzki plane (or Nemytskii plane, Nemytskii's tangent disk topology), is a topological space. It is a completely regular Hausdorff space (also called Tychonoff space) that is not normal. It is named after Robert Lee Moore and Viktor Vladimirovich Nemytskii.
Let $\Gamma=\{(x,y)\in \mathbb{R}^2:y\geq 0\}$ and define a topology on $\Gamma$ given by the local basis
$$B(p,q)= \begin{cases} \{(x,y):(x-p)^2+(y-q)^2<\epsilon^2:\epsilon>0\},&\text{ if }q>0 \\ \{(p,0)\}\cup\{(x,y):(x-p)^2+(y-\epsilon)^2<\epsilon^2:\epsilon>0\},&\text{ if }q=0 \end{cases}$$
Thus the subspace topology inherited by $\Gamma \setminus \{(x,0):x\in \mathbb{R}\}$ is the same as the subspace topology inherited from the standard topology of the Euclidean plane.