TITLE: Examples for subspace of a normal space which is not normal
QUESTION [22 upvotes]: Are there any simple examples of subspaces of a normal space which are not normal? 

I know closed subspace of a normal space is normal, but open subspace in most cases which I can think of are also normal.

REPLY [7 votes]: Let, $X= \{a,b,c,d\}$
And $T= \{ \emptyset ,X,\{d\} , \{b,d\} ,\{ c,d\},\{b,c,d\}\}$
Then $(X,T)$ is a topological space.
Since $(X,T)$ has no pair of disjoint non-empty closed sets, $(X,T)$ is a normal space.
Consider ,$Y=\{b,c,d\}$ of $X$.
Then $T(Y)= \{\emptyset,Y, \{d\} , \{b,d\} ,\{ c,d\}\}$
Then $\{b\}$ and $\{c\}$ are disjoint closed sets in $(Y,T(Y))$ and they cannot be separated in $(Y,T(Y))$.
Hence ,$(Y,T(Y))$ is not a normal space.