The excerpt is from "Topology Without Tears" by Sidney Morris. I am not able to get the rationale for the steps (1) and (2) in the above picture. Why does "[0,1] = [0,1]$\cap$Y " imply that [0,1] is closed in (Y,$T_1$) ? And why "[0,1]=(-1,1.5)$\cap$Y" imply [0,1] is open in (Y,$T_1$) ?
My reasoning for why [0,1] is clopen is as follows. Since (Y,$T_1$) = [0,1]$\cup$[2,3], by definition [0,1] is in this topology, hence it is open. Since it is the complement of [2,3] in Y, it is closed. Why taking an intersection with arbitrary sets is required as in the above picture? What have I misunderstood here?