1. Consider the Nemiytzki's Tangent Disk topology discussed in class. This topology is defined on the closed upper half plance H = P U L, where 

• P ={(x,y)  in R^2/ y > 0}
• L={(x,y) in R^2/ y = 0}

 A basis for this topology is given by the following sets

Open sets of P in the standard topology of R^2.
All subsets of X  of the form D U {(x,0)}, where D is a disk in P tangent to  L at the point (x,0)

1. Prove that this space is separable, T0, T1, T2, T2 1/2, T3 and T3 1/2
2. Prove that this space is not second countable.
3. Extra credit (not mandatory) Prove that it is not T4.