@HenryT.Horton TO prove $\operatorname{int}(A)$, $\operatorname{int}(X\setminus A)$ and $\partial A$ are mutually disjoint, and then prove $\operatorname{int}(A)\cup\operatorname{int}(X\setminus A)\cup\partial A=X$,
You want a continuous map $F: (\Bbb R^n \setminus \{0\}) \times [0,1] \longrightarrow \Bbb R^n \setminus \{0\}$ such that $F(x,0) = x$ and $F(x,1) \in S^{n-1}$ for all $x \in \Bbb R^n \setminus \{0\}$ and $F(a,1) = a$ for all $a \in S^{n-1}$
If $x \in D^{n+1}$, then $x/\|x\| \in D^{n+1}$ and since your homotopy was a convex combination of $x$ and $x/\|x\|$ and $D^{n+1}$ is convex, the homotopy still works
He says "Since (the function $f$ under consideration) is analytic and not constant, it has finitely many zeros in a compact set. So any plane has only finitely many intersections with (the graph of $f$)."
I do not understand what is being concluded in the first sentence. Does he mean that $f$ has only finitely many zeroes overall, or just that given some compact set $C$ it has only finitely many zeroes in $C$?
it means for any $a,b$, the combo $af_1+bf_2$ is not the zero map. this doesn't mean it never takes on a zero value, as nonzero maps can have zeros. nawatImean?
I believe that exercises on this topic can be found in many introductory texts on discrete mathematics (for finite sums) or calculus (for both finite in infinite sums).
@Chris'sister I have some doubts about possibilities of doing research on multiple sums. (It is pretty vague description of what the research should be about.)
But maybe summability methods for double sequences are close enough to this and it would be possible to make some research in this area. (Not sure to which extent one can expect some useful results.) I'll list a few papers I am aware of. Most of them just randomly caught my attention in the past and I make a note; it's more a sample than a suggestion where study of this are could start.
Boos, Leiger, Zeller: Consistency theory for SM-methods, Acta Mathematica Hungarica, 1997; doi: 10.1007/BF02907056
Zeltser: On Conservative Matrix Methods for Double Sequence Spaces, Acta Mathematica Hungarica, doi: 10.1023/A:1015636905885
Patterson: Analogues of some fundamental theorems of summability theory, link
J. D. Hill: Almost-convergent double sequences; Tohoku Mathematical Journal, Volume 17, Number 2 (1965), 105-116. link
Perhaps you'll get some other suggestions from other users.
And perhaps you should clarify what you mean under being interested in new research on this topic. Do you plan to write some kind of project/essay/thesis and you want choose an area which is interesting for you?
@draks The term I am accustomed to is that the exponential integral is an antilimit for that series. The series itself is divergent (being an asymptotic series), but 1. it can be made sensible, and 2. it can be analytically continued.
@MarkDominus Have you seen Martin's survey in meta?
@draks As I said: the series is not convergent in itself, but a certain value can be associated to it via, say, analytic continuation of an equivalent form. That value is the antilimit.
