Observable sets

It is important to know the normal world. Arts and sciences are complicated, but we need to perceive the common sense. I will describe it mathematically.

1=objects, 2=devices or machines (eyes), 3=observers (brain)

This case is very complicated, and it is almost artificial intelligence, according to John von Neumann.
At first, our eyes are not devices like Google Glass.
When you see the objects, the information is sent to your brain. If your brain were broken like me, this must be disorder.

Math isn't chaos.

However, your eyes and brain are connected, so your perception may be sometimes different from others. You are the subject, not objects like devices. You may say that reality is abstract. You are also egocentric. This is our existential problem. You may be the transcendence like Buddha, and you are myself and there is no boundary in this world. I see and you also see. This is integration, no duality.

Math isn't religion.

We share common knowledge. This is commutative.

	A	B	C	D	E	F	G
A	○	×	○	○	○	○	○
B	×	○	×	○	○	○	○
C	○	×	○	○	○	○	○
D	○	○	○	○	○	○	○
E	○	○	○	○	○	×	○
F	○	○	○	○	×	○	○
G	○	○	○	○	○	○	○

[X,Y]=XY-YX≠0→×=noncommutative

[X,Y]=XY-YX=0→○=commutative

A and B are noncommutative, and B and C are also noncommutative. They are linked.

S={A,B,C,D}

S and S' are commutative. A-B-C is noncommutative, so it isn't S'. D and G are commutative, so they are S'. E-F is noncommutative, but D and G are commutative. Therefore, it is S'.

S'={D,E,F,G}

S'' is double commutative, so it includes D and G.

D is the center, but G connect S and S'.

D and G are commutative, so S and S'' are commutative.

S=S''

Quaternion

Prime numbers include the imaginary space in two dimensions. William Rowan Hamilton has found the quaternions. It extends complex numbers.

H is the sum of quaternions.
R is space of numerical vectors.

H={(a,b,c,d)|a,b,c,d∈R}
H=a+bi+cj+dk

i^2=j^2=k^2=ijk=-1

-1=ijk
-i=i^2jk
-i=-jk
∴i=jk

×	1	i	j	k
1	1	i	j	k
i	i	-1	k	-j
j	j	-k	-1	i
k	k	j	-i	-1

∴
ij=k, ji=-k, jk=i, kj=-i, ki=j, ik=-j

ij≠ji

H=a+bi+cj+dk (i=jk,j=ki,k=ij)

You are dazing.

Hausdorff space

It is hard to define emptiness. Hausdorff space is known as empty set Φ.



X is a topological space.

X=(x,y)

x and y are dots.

U∩V=Φ

U is vicinities of x.
V is vicinities of y.

X×X→Δ={(x,x)|x∈X} is closed.