Topology: Lecture 1: adherence in Euclidean Space

[cobovega]

Alright!! You're back! As long as it remains reasonably viable, and of course, I would presume, enjoyable to you, keep doing what you're doing: I can't tell you how helpful your videos have been, Professor Cook. Thank you.

[realmartinshkreli]

This is priceless stuff, thank you for making these videos. Meanwhile a guy who says 'skrrt' has 400, 000, 000 views. What a world we live in. Quick maths, indeed.

[audxc]

I was wondering about the closed set Defn but I figured manetti was trying to just say a set is closed if its complement is open. Just in 'adherent' language. Also if you think about constructing a circle as a simplex it is clear a line and a point are homeomorphic to a circle.

[xoppa]

4:19 From Manetti's definition of an adherent point of A, "x adheres to A iff for any δ > 0 there exists y ∈ A such that d( y, x ) < δ", it follows that an isolated point should be an adherent point. In the video you seem to be arguing that the isolated point P in A , where A = A_1 U {P} U A_2, is not an adherent point of A.
For example 3 is an adherent point of the set (1, 2) U {3}, because for any δ > 0 there exists y ∈ A such that d(y, 3) < δ, namely choose y = 3. Then d(y, 3) = d(3, 3) = 0 < δ is true.


"The limit points of a set should not be confused with adherent points (also called points of closure) for which every neighbourhood of x contains a point of S (that is, any point belonging to closure of the set). Unlike for limit points, an adherent point of S may be x itself. A limit point can be characterized as an adherent point that is not an isolated point. "

Definitions are quite tricky.