Normal space need not be completely regular I know that a $T_4$ space is Tychonoff space. But a Normal space need not be completely regular. Give me a counterexmple in support of that.  
 A: Take for example the Sierpiński space $X=\{a,b\}$ with the topology $\{X,\emptyset, \{a\}\}$. The closed set $\{b\}$ and the point $a$ cannot be separated by disjoint open sets as the only open set containing $\{b\}$ is $X$. Since there are no disjoint non-empty closed sets, $X$ is automatically normal.
A: (This is more of a comment than an answer, but since I don't have enough reputation for adding comments yet, I'm writing it as an answer.)
Whether all normal spaces are completely regular or not depends upon what definition of "completely regular" you are assuming. Some authors (e.g., Cullen 1968, p. 130; Munkres - Topology, p. 211) require the space to be a $T_{1}$ space in order to be a completely regular space. 
So if you assume that the singletons are closed, then every normal space is completely regular. If you don't, then you can construct a topological space like the one described in Stefan's answer, in order to get a normal space which isn't completely regular.
A: This is not possible because by Urysohn's lemma you can consider a point to be a closed set, then we get completely regular space.
So every normal space is completely regular space.
(See in TOPOLOGY by James R. Munkres)
