Is the punctured plane a normal space? Is a normal space without a point normal?

Question

At first I thought is was obvious that $\mathbb R^2 \setminus \{ P \}$ is normal as a topological subspace of $\mathbb R^2$. But what about if we have two closed sets in $\mathbb R^2$, $F$ and $E$, such that $F \cap E = \{ P \}$. How can we show in general that we can separate $F \setminus \{ P \}$ from $E \setminus \{ P \}$ with open sets in $\mathbb R^2 \setminus \{ P \}$? Is this even true?

To generalise, is it true in general that the subspace obtained from a normal space by removing a point stays normal?

Answer 1

Hint: all metric spaces are normal spaces.

Answer 2

No, in general remvoing a point from a normal space can make the resulting subspace non-normal. The classic case is the Tychonoff plank, the subspace

$((\omega+1) \times (\omega_1+1)) \setminus \{(\omega, \omega_1)\}$ of the compact Hausdorff (hence normal) space $(\omega+1) \times (\omega_1+1)$. (Each factor in the order topology).

But for metric and ordered spaces normality is hereditary. So for the plane it's no issue.