I have this question here from a course in general topology course I could not answer all of so I am asking here, it reads:
Let $ (X,\tau) $ be a $ T_4 $ topological space (normal Hausdorff) with $ |X| = \infty $
a. We are to define the definition of a $ T_4 $ topological space. Ok I know what the definition of this space is.
b. We are to prove the existence of disjoint open non empty sets, U and V with U an infinite set.
c. We are to prove in X there exists an infinite sequence of disjoint open sets (non empty).
d. We are to prove the space of continuous bounded real valued functions $ C_b(X,R) $ with the supremum norm $ L^{\infty} $ is not locally compact.
Ok so what I have tried to do in part b was try to find an infinite closed set and with another singleton a closed set we can certainly separate them with disjoint open sets due to normality of X but besides singletons and finite sets I do not know of closed sets... On part c I am stuck as I could not construct such a sequence and part d is a complete mystery to me. I thank you for all the help