Topological Space Theory is the most difficult field for mathematics students
◇◇◇◇◇◇Mathematical Supplements◇◇◇◇◇◇
In the video, in order to make the explanation easier to understand, the rigor of modern mathematics has been removed, or there are parts that I would like to explain a little more but have omitted because they would go off on a tangent. Therefore, I have tried to add additional details in the explanation section to the parts that I am concerned about. :
(Supplement 1) About the word "density"
Density is a term that does not appear in the basic parts of set theory, but I use it in the video to make it easier to imagine.
The word density contains the image of "points packed tightly together," but if you try to realize this image mathematically, you would actually need structures such as distance and topology.
Concepts that can be imagined as density on the real line include "concentration of a set," "density," "sparse set," "skinny set," "measure," and "continuity." Roughly speaking, we can have the intuition that a high cardinality means that it tends to become locally dense, and a sparse set has a small measure and low cardinality, but mathematically these concepts are not neatly related.
For example, the Cantor set is a sparse set with a one-dimensional Lebesgue measure of 0, but has continuous cardinality, and Q is dense on R, but its one-dimensional Lebesgue measure is 0 and countable. The even thicker Cantor set is an example of a set that has a positive Lebesgue measure but is sparse.
In this way, when discussing whether points are densely packed or not, it is necessary to decide which concept is being adopted.
(Supplementary Note 2) Density intermediate between Z and Q
In the video, a derived set is used to define density. A set that will eventually become an empty set through derived set operations is imagined as "density intermediate between Z and Q", but there is no mathematical reason why it is intermediate between Z and Q.
In terms of Cantor's research, the Cantor set is raised as a uniqueness set, so it may be more natural to express it as an intermediate density between Z and the Cantor set.
(Supplementary Note 3) Definition of an accumulation point
In a Euclidean space, the fact that x is an accumulation point of a subset A can be expressed in the following two ways:
(a) There are an infinite number of points of A in any neighborhood of x: ∀U∈N(x), |A∩U|≧ℵ₀
(b) There are points of A other than x in any neighborhood of x: ∀U∈N(x), U∩(A∖{x})≠∅
If you want to emphasize that it is a dense point, it is natural to define (a) as an accumulation point. Indeed, (a) and (b) mean the same thing in Euclidean space, but they are different in general topological spaces.
In a topological space, (b) is the definition of an accumulation point, and when (a) holds, x is called an ω-accumulation point.
For example, in a space (R,T) where R has a coherent topology T, these concepts are different. In fact, for example, for A={0,1}⊂R, any point of R in (R,T) is an accumulation point of A, but it is not an ω-accumulation point.
(Supplementary Note 4) About ordinals
In the video, I said that ordinals are "like a sequence of ω with operations applied to it," but this is just an explanation to give a simple image of ordinals, and it does not define ordinals.
Ordinals are defined intrinsically in the language of set theory, and it is possible to think of ordinals that cannot be expressed using arithmetic operations or powers for ordinals below ω.
◇◇◇◇◇◇◇Video Errata◇◇◇◇◇◇
11:25
Incorrect: ∃M[ ∅∈M ∧ ∀x[ x∈M → x∪{x}∈ "I" ] ]
Correct: ∃M[ ∅∈M ∧ ∀x[ x∈M → x∪{x}∈ "M" ] ]
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BGM: "Atelier and the Cyber World" Sharou
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2024/08/14: Re-uploaded due to audio issues