This question is from Wayne Patty's Section 5.3
Let $T=${$U \in P(\mathbb{R}): 0 \notin U$ or $ \mathbb{R} \setminus U $ is finite } is (a) completely normal (b) not perfectly normal.
(a) A $T_1$- space (X,T) is completely normal provided that whenever A and B are subsets of X such that $A\bigcap \overline{B}= \overline{A} \bigcap B =\emptyset$, there are disjoint open sets U and V such that $A\subseteq U$ and $B\subseteq V$.
Let A and B be the subsets of $\mathbb{R}$ satisfying $A\bigcap \overline{B}= \overline{A} \bigcap B =\emptyset$ then I don't understand which result I should use to show the existence of disjoint open sets U and V.
(b) A $T_1$ space (X,T) is perfectly normal provided that for each pair of disjoint closed subsets C and D in X there is a continuous function $f:X \to I$ such that $C= f^{-1}(0)$ and $D= f^{-1}(1)$.
Now, Let C and D be disjoint closed susbsets of $\mathbb{R}$ we have $C^c$ and $D^c$ are open which means that either $0\notin C^c$ or $\mathbb{R} -C^c$ is finite and $0\notin D^c$ or $\mathbb{R} - D^c$ is finite.
But I am not able to construct such a function f. Can you please help with that?
For a) show $X$ compact and Hausdorff e.g. Every subspace of it is either discrete (if it misses $0$) or again compact, so normal in all cases. Hence $X$ is completely normal (which is equivalent to every subspace being normal).
For $b)$ show that $\{0\}$ is closed but not a $G_\delta$.