First of all - many thanks for the series! So far it's great 🙂 I'm really confused about the boundary of C (at the end of the video) - why isn't it ∂C={1,2}? Shouldn't boundary points be non-dependent on membership of the set?
A point is a boundary point if, for any epsilon > 0, epsilon ball of that point has a non-zero intersection with both the set and it's complement. Epsilon ball of 1: Bₑ(1) = (1; 1 + e) for e ⩽ 2, and Bₑ(1) ⊂ C for e ⩽ 1, so Bₑ(1) ∩ Cᶜ = ⵁ, so that doesn't work
![Functional Analysis 4 | Sequences, Limits and Closed Sets [dark version]](http://i.ytimg.com/vi/UHWKPqzFVYI/mqdefault.jpg)
![Functional Analysis 4 | Sequences, Limits and Closed Sets [dark version]](/img/tr.png)
![Functional Analysis 16 | Compact Sets [dark version]](/img/n.gif)
I wonder why this video doesn't get so many views and likes
It's because it's maths :D
Because it's the corrected (dark ;-D) version of the brighter yellow version that came first (both are identical other than colors).
Hint for future me: (b) at (9:11) says A is also closed -- 1 isn't in X. Also relevant for (c).
First of all - many thanks for the series! So far it's great 🙂
I'm really confused about the boundary of C (at the end of the video) - why isn't it ∂C={1,2}? Shouldn't boundary points be non-dependent on membership of the set?
The boundary points are all the points that lie on the boundary between inside and outside. So it depends on the chosen topology.
A point is a boundary point if, for any epsilon > 0, epsilon ball of that point has a non-zero intersection with both the set and it's complement.
Epsilon ball of 1:
Bₑ(1) = (1; 1 + e) for e ⩽ 2,
and
Bₑ(1) ⊂ C for e ⩽ 1,
so
Bₑ(1) ∩ Cᶜ = ⵁ, so that doesn't work
Sir did you make videos about fuzzy set if you makes on that please give me the link