The Dover edition of Mendelson contains hints and answers to selected problems, but not full solutions. For example, it might say: "A set is closed if its complement is open." That’s a hint, not a solution. You need more.
Master Topology with Bert Mendelson: A Guide to the Text and Its Solutions Bert Mendelson’s Introduction to Topology Introduction To Topology Mendelson Solutions
Show that the discrete metric ( d(x,y) = 0 ) if ( x=y ), else 1, induces the discrete topology. The Dover edition of Mendelson contains hints and