1. ## Theorem on Closure

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

3)

4)

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.

2. The essence of the problems you posted is understanding definitions.

A point is interior to a set if some open set containing the point is a subset of the set. An intuitive way to look at it is: a point is interior if every point close to it is also interior. Note too the existential, some open set.

On the other hand, a point is a boundary point if every open set containing the point also contains a point in the set and a point not in the set. That is: every open set about the point intersects both the set and its complement.

Therefore, no interior point can be a boundary point and visa versa.
Thus it is easy to apply definitions to prove
$\begin{gathered}
2)~~ \beta (S) = \overline S \backslash S^o \hfill \\ \\
3)~~\beta (S) = \overline S \cap \overline {S^c } \hfill \\
\end{gathered}$

You may also learn some things by viewing the LaTeX.
It really is best to learn to code.