This is prob 7 on pg 56 of "Taylor, General Theory of Functions and Integration, First Ed, 1965"
Prove 3). Hint: Prove 4) and observe that when this is combined with 2) we get 3).
1) Definition of Closure. S' is the set of all accumulation points of S
2) Definition of boundary of S
S0 is interior of S
COMMENT. I like the style, level, and brevity of the first few chapters. I just want a feel for fundametals of analysis, like Heine-Borel. But the author is constantly adding theorems to be proved by reader, without answers. This drives me nuts. I give the above as a sample problem. I don't even know where to start. All I can do is give words to the formulas and show they make intuitive sense.