Well, only changing one element in the sequence which doesn't make it so that $x_{2n}=0$ for all $n\in\Bbb{N}$ will keep it outside of $X$, but I feel unsure about how distances work here.
@TedShifrin Well they have a lot of PDE courses which also include functional analysis, but I would be more interested in something like a course following Conway's book than one doing PDEs (Also the modules handbook is super vague, I'll ask more infos when I'm there)
Can someone explain me what's global analysis? All I know is that it's "analysis on manifolds", but what are some important or deep results in this field for example?
Write down the definition for $\|(w_n)-(z_n)\|_1<\epsilon$.
A big one is the Calabi conjecture (proved by Yau in the late 70s), @Alessandro. Google.
Lots of stuff that Eric probably can tell you that his adviser and colleagues have done with minimal surface theory in the past decades.
@Oskar: If you want to picture things a bit, draw a horizontal number line, and at each integer $n\ge 1$ draw a vertical0 line in the plane. You can plot your sequence by picking a point on each of those vertical lines. Now how do you measure distance between two such?
Every point $x_1,x_2,\dots$ in the sequence $(x_n)$ would have a distance to the points $y_1,y_2,\dots$ in $(y_n)$. The distance between the sequences as a whole seems a bit abstract to me right now.
That $\epsilon$ must be smaller than the difference from all $x_{2n}\neq0$. Otherwise, we could find a sequence which has $x_{2n}=0$ for all $n\in\Bbb{N}$ and thus be inside $X$.
Right, choosing $0<\epsilon\le |w_{2s}|$ does it, @Oskar, because even if we use up all our $\sum |w_n-y_n|$ in that one slot, we don't have enough room to go down to $0$.
If something hasn't popped up before, and Wikipedia doesn't have a good article about it, then it's kind of hard to find a comprehensive source for the material.
There are lots of other books to look at, @Oskar. My library is depleted so I can't recommend too many specifically. Simmons has a nice book on analysis and topology that's quite readable.
Okay so let's say we have $\mathbb{RP}^n$ as $S^n$ mod antipodal points, I want to see why this is $D^n$ with antipodal points of the boundary identified.
Okay that makes sense actually, so then you take a point from each equivalence class in $S^n$ except you take both points on the equator/boundary of $D^n$, so you identify them
A point in $S^n/{(p\sim-p}$ is a pair of opposite points. Either both points are on the equator, or one is in the open southern hemisphere and the other is in the open northern hemisphere
1) If 1 is open, then the group is discrete. Assume not. 2) Your subgroup is discrete with the subspace topology, so there is a neighborhood of 1 in the larger group that only intersects 1 in that subgroup. 3) By non-discreteness, 1 is not open. So every open neighborhood has other elements of G. 4) You cannot accumulate to those other elements, as the only element of your subgroup in a small neighborhood is 1!
I recognize I did not clearly delineate between the two above
(1) and (3) are about the large group. (2) about the subgroup. (4) about a neighborhood of the identity in the large group that has only the identity of the small group
