Math 5587 – Lecture 15 Jeff Calder
نویسنده
چکیده
1.1 Topology Let us first recall some basic Euclidean topology. For x0 = (x0, y0, z0) ∈ R3 and r > 0 we define the open ball of radius r > 0 centered at x0 to be B(x0, r) = {x ∈ R | ‖x− x0‖ < r}. Recall that ‖x‖ = √ x2 + y2 + z2 is the Euclidean norm, or length, of the vector x. Definition 1 (Open set). We say a set D ⊂ R3 is open if for every x0 ∈ D, there exists a radius r > 0 such that B(x0, r) ⊂ D. In other words, a set D is open if every point x0 ∈ D can be moved by a tiny amount in any direction and still remain in D. The open ball B(x0, r) is open. Definition 2 (Closure). Given a set D ⊂ R3, the closure of D, denoted D is defined by D = {x ∈ R | x = lim n→∞ xn for some sequence xn ∈ D}.
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