talking about function and set graphs... again

• we can see the "drawing" of a compact set E a finite cover of E with neighborhoods of finitely many elements of E ; meaning, there are Ω₁, ..., Ω_n open neighborhoods of e₁, ..., e_n respectively, with finite n, such that Ω₁ ∪ ... ∪ Ω_n = E.
• another more general characterisation, is that a "drawing" of E contains a fibre of E with balls ; basically, that means you can put open balls around all points of E so that it stays in the drawing. despite the possible infinitude de points in this version, these two definitions coincide, at least with finitely many dimensions, probably in separable spaces too but i'm too lazy to check (and more generally, i have too little intuition to take an informed guess).
• we say that E is dense in F if cl(E) = F, which for E⊆F is graphically characterized by E having the same drawings as F : all drawings of F also work as drawings of E. this representation allows us to show that if E is dense in F or vice-versa, then E and F are both dense in E∪F, as they share the same drawings, that E⊆E∪F, and that F⊆E∪F.

1. let's take an injection φ : E → F ⊆ [0, 1] with E⊆ℝⁿ the set we want to graph.
2. using E⊆ℝⁿ, we can graph E using the image of E by (cos(πk/n − π/(2n))·φ, φ, sin(πk/n − π/(2n))·φ)_{k ∈ {1..n}} if E is n-dimensional with finite n. otherwise, if E is a part of H an infinitely dimensional hilbert space of cardinal at most #ℝ, we can use the image of E by (cos(α∘⟨·, b⟩/2)·φ, φ, sin(α∘⟨·, b⟩/2)·φ)_b∈BH, where B_H is a hilbertian basis of H, an arbitrary α : 𝕂 → [0,2π].