Let $X$ and $X'$ denote the same set under different topologies $\tau $ and $\tau'$ where $\tau' \supset \tau$. I have to decide what one of them would be like if the other one is like Regular or Normal spaces.
So , first I consider $X=\mathbb R=X'$ and $\tau'$=euclidean topology and $\tau$= co-finite topology
Then easily $X'$ is both normal and regular but $X$ is neither.
Going the other way round is a little problem. For the given condition tells that any open set in $\tau$ is open in $\tau'$ but the converse may not hold.
Now if I assume that $X$ is Regular. If $p'$ is a point in $X'$ disjoint from the closed set $C'.$ If I want to prove that $X'$ is Regular then I have to find open sets $U'$ and $V'$ in $X'$ such that $$U'\cap V'=\varnothing$$ and $$p'\in U' \\C'\subset V'$$ . This would be done if only I could tell that $C=C'$[in $X$] were closed in $X$.
How can I , if at all , prove that $?$ Or if that is not the fact then I need some counter-example to disprove that statement .
Please help me with the $Regular$ , I will try to do the $Normal$ on my own .