Question (purpose: finding out if I understood what you're saying): Suppose my set X is a collection of socks. I associate a "coordinate" system for a sock by providing its quantitative features, such as y=(age, color, size, number_of_holes, ...) as y = f(x), where x is a sock and y is that tuple. I make these n features continuous and associate real numbers with them (*) so they can take any value in a compact subset of R^n. According to what you said, I should then I be able to create a topology over socks, merely by giving them coordinates, right? I was thinking that I could specify an epsilon ball in the feature space, and then find as the inverse image of that environment a bunch of socks in the sock pile ( or perhaps no sock at all: The empty set of socks got to be in the set of socks). Does this mean I have now a topology on my sock collection? Apparently all I need to do is sort them and lay them out nicely so I can put coordinates on them. Right or wrong? (I know it's a silly example. Sorry :-) (*) colors can be quantified by an optical spectrum.
Subspace topology is dual to quotient topology. Union is dual to intersection. Conjunction is dual to disjunction. Homology is dual to co-homology. Elliptic curves are dual to modular forms -- Fermat's last theorem. "Always two there are" -- Yoda.
You are a born communicator. I am interested in attending more of your lectures on Topology. Do you have a set of notes in pdf online? Thank you very much
@@DanielChanMaths Equivalence, similarity = duality. Gravitation is equivalent or dual to acceleration -- Einstein's happiest thought, the principle of equivalence (duality). Energy is dual to mass -- Einstein. Subspace topology is dual to quotient topology. Syntropy (prediction) is dual to increasing entropy -- the 4th law of thermodynamics! "Always two there are" -- Yoda.
Tell me soluation for Let M be a closed linear subspace of a normed linear space N. If the norm of a coset x + M in the quotient space N/M is defined by || M || inf {|| ||: M}, x x m m + = + ∈ then prove that N/M is a normed linear space. Further prove that if N is a Banach space then so is N/M
Equivalence or similarity = duality. Duality: "Two equivalent descriptions of the same thing" -- Leonard Susskind, physicist. Self similarity = fractals or self duality. In physics entropy or "the arrow of time" does not have an inverse function, entropy is not bi-continuous or bijective. Potential energy is dual to kinetic energy, apples fall to the ground because they are conserving duality. Thesis is dual to anti-thesis creates converging thesis or synthesis -- the time independent Hegelian dialectic. Injective is dual to surjective synthesizes bijective. Entropy is a not a bijective function! The future is dual to the past -- time duality. Space is dual to time -- Einstein.
@@biblebot3947 Try and learn a couple of new things everyday. There is a actually a 4th law of thermodynamics in physics:- Syntropy (prediction, projection) is dual to increasing entropy. Your mind converts entropy or information into syntropy which is used to make optimized predictions about the world. Concepts are dual to percepts -- the mind duality of Immanuel Kant. Perceptions which are expectations or predictions in neuroscience are converted into conceptions via rational, analytic thinking -- thinking is a syntropic process. "Through imagination and reason we turn experience into foresight (prediction)" -- Spinoza describing syntropy. Deductive inference (rational, analytic) is dual to inductive inference (empirical, synthetic) -- Immanuel Kant. Inferences or predictions are dual (deductive, inductive). Mathematics involves deductive reasoning but physics involves inductive reasoning. The two are mutually exclusive or dual but end up with the same result. Energy is duality, duality is energy in physics -- once you realize this it is all down hill from here, this is why a 4th law of thermodynamics is important. Everything is made from energy in physics or there is an underlying duality to everything. "Always two there are" -- Yoda. Perpendicularity or orthogonality = duality in universal hyperbolic geometry. Watch the following videos:- czcams.com/video/EvP8VtyhzXs/video.html The Christian cross is composed of two perpendicular lines = duality. The point duality theorem is dual to the line duality theorem -- points are dual to lines, the principle of duality in projective geometry. Waves are dual to particles -- quantum duality. Study duality as a real physical quantity --- fundamental physics. Good news is dual to bad news. The bad news is:- "Philosophy is dead" -- Stephen Hawking. You cannot use philosophy or metaphysics according to Stephen Hawking, is this true?
@@biblebot3947 The conservation of duality (energy) will be known as the 5th law of thermodynamics -- Generalized duality, but first you have to accept the 4th law.
@@biblebot3947 Yes what exactly is duality? It is an ongoing question, questions are dual to answers. Mathematicians have a different definition to physicists -- see my first comment or Leonard Susskind's definition. Duality is literally energy in physics, they are the same thing. In a gravitational field potential energy is converted into kinetic energy -- duality is always conserved here. Waves are dual to particles -- pure energy or light is dual -- quantum duality. Thesis is dual to anti-thesis creates the converging thesis or synthesis -- the time independent Hegelian dialectic. God (thesis) is dual to the Christ consciousness (anti-thesis) synthesizes the holy spirit or mind/soul -- Hegel. The council of Nicaea in 325 AD was set up to answer the question, is the Christ consciousness the same substance as God or a different substance? Same is dual to different, homo is dual to hetero. Christianity is based upon duality and not triality or trinity! Mind (the internal soul, syntropy) is dual to matter (the external soul, entropy) -- Descartes. According to Descartes the mind/soul is dual -- self duality. Duality: two sides of the same coin (heads is dual to tails). Optimism is dual to pessimism. The optimist says the glass of water is half full, the pessimist says the glass of water is half empty -- two dual perspectives of the same thing.
Very good explanantions, thank you very much for the effort.
thanks very much for the illustration, i think that i proceed a bit more
Question (purpose: finding out if I understood what you're saying): Suppose my set X is a collection of socks. I associate a "coordinate" system for a sock by providing its quantitative features, such as y=(age, color, size, number_of_holes, ...) as y = f(x), where x is a sock and y is that tuple. I make these n features continuous and associate real numbers with them (*) so they can take any value in a compact subset of R^n. According to what you said, I should then I be able to create a topology over socks, merely by giving them coordinates, right? I was thinking that I could specify an epsilon ball in the feature space, and then find as the inverse image of that environment a bunch of socks in the sock pile ( or perhaps no sock at all: The empty set of socks got to be in the set of socks). Does this mean I have now a topology on my sock collection? Apparently all I need to do is sort them and lay them out nicely so I can put coordinates on them. Right or wrong?
(I know it's a silly example. Sorry :-)
(*) colors can be quantified by an optical spectrum.
Why do they use pi notation in π: X->Y in algebraic topology?
It's usually used for maps which are like projections. I guess rather than using p for the projection, they like to use pi since p is often a point.
DanielChanMaths
Oh I see thank you
Subspace topology is dual to quotient topology.
Union is dual to intersection.
Conjunction is dual to disjunction.
Homology is dual to co-homology.
Elliptic curves are dual to modular forms -- Fermat's last theorem.
"Always two there are" -- Yoda.
You are a born communicator. I am interested in attending more of your lectures on Topology. Do you have a set of notes in pdf online? Thank you very much
Unfortunately, there are only videos, no accompanying notes.
DanielChanMaths an advanced mathematical textbook you would suggest, then? :)
@@DanielChanMaths Equivalence, similarity = duality.
Gravitation is equivalent or dual to acceleration -- Einstein's happiest thought, the principle of equivalence (duality).
Energy is dual to mass -- Einstein.
Subspace topology is dual to quotient topology.
Syntropy (prediction) is dual to increasing entropy -- the 4th law of thermodynamics!
"Always two there are" -- Yoda.
What is S1? I think Sphere 1?
I think my confusion is based on notations
S with a superscript n means the n-dimensional sphere so S^1 is just the circle. The notation is fairly standard.
*n+1 dimensional
Can you give me a book where I can study this subject?
Munkre's Topology book is considered holy on general topology.
For something lighter, you can also look at McCarty's "Topology".
My personal bible on this subject is a book from 1970 called Topology for Analysis by Albert Wilansky.
there is a nice free book online called topology without tears
Bert Mendelson's intro to topology is a good one for an introduction to the subject.
Tell me soluation for Let M be a closed linear subspace of a normed linear space N. If the norm of a
coset x + M in the quotient space N/M is defined by
|| M || inf {|| ||: M}, x x m m + = + ∈
then prove that N/M is a normed linear space. Further prove that if N is a Banach
space then so is N/M
Equivalence or similarity = duality.
Duality: "Two equivalent descriptions of the same thing" -- Leonard Susskind, physicist.
Self similarity = fractals or self duality.
In physics entropy or "the arrow of time" does not have an inverse function, entropy is not bi-continuous or bijective.
Potential energy is dual to kinetic energy, apples fall to the ground because they are conserving duality.
Thesis is dual to anti-thesis creates converging thesis or synthesis -- the time independent Hegelian dialectic.
Injective is dual to surjective synthesizes bijective.
Entropy is a not a bijective function!
The future is dual to the past -- time duality.
Space is dual to time -- Einstein.
Where do you learn this stuff?
@@biblebot3947 Try and learn a couple of new things everyday.
There is a actually a 4th law of thermodynamics in physics:-
Syntropy (prediction, projection) is dual to increasing entropy.
Your mind converts entropy or information into syntropy which is used to make optimized predictions about the world.
Concepts are dual to percepts -- the mind duality of Immanuel Kant.
Perceptions which are expectations or predictions in neuroscience are converted into conceptions via rational, analytic thinking -- thinking is a syntropic process.
"Through imagination and reason we turn experience into foresight (prediction)" -- Spinoza describing syntropy.
Deductive inference (rational, analytic) is dual to inductive inference (empirical, synthetic) -- Immanuel Kant.
Inferences or predictions are dual (deductive, inductive).
Mathematics involves deductive reasoning but physics involves inductive reasoning.
The two are mutually exclusive or dual but end up with the same result.
Energy is duality, duality is energy in physics -- once you realize this it is all down hill from here, this is why a 4th law of thermodynamics is important.
Everything is made from energy in physics or there is an underlying duality to everything.
"Always two there are" -- Yoda.
Perpendicularity or orthogonality = duality in universal hyperbolic geometry.
Watch the following videos:-
czcams.com/video/EvP8VtyhzXs/video.html
The Christian cross is composed of two perpendicular lines = duality.
The point duality theorem is dual to the line duality theorem -- points are dual to lines, the principle of duality in projective geometry.
Waves are dual to particles -- quantum duality.
Study duality as a real physical quantity --- fundamental physics.
Good news is dual to bad news. The bad news is:-
"Philosophy is dead" -- Stephen Hawking.
You cannot use philosophy or metaphysics according to Stephen Hawking, is this true?
@@biblebot3947 The conservation of duality (energy) will be known as the 5th law of thermodynamics -- Generalized duality, but first you have to accept the 4th law.
@@hyperduality2838 what exactly is this “duality” that’s so important?
I get a vague sense of what it is, but is there a rigorous definition of it?
@@biblebot3947 Yes what exactly is duality? It is an ongoing question, questions are dual to answers.
Mathematicians have a different definition to physicists -- see my first comment or Leonard Susskind's definition.
Duality is literally energy in physics, they are the same thing.
In a gravitational field potential energy is converted into kinetic energy -- duality is always conserved here.
Waves are dual to particles -- pure energy or light is dual -- quantum duality.
Thesis is dual to anti-thesis creates the converging thesis or synthesis -- the time independent Hegelian dialectic.
God (thesis) is dual to the Christ consciousness (anti-thesis) synthesizes the holy spirit or mind/soul -- Hegel.
The council of Nicaea in 325 AD was set up to answer the question, is the Christ consciousness the same substance as God or a different substance?
Same is dual to different, homo is dual to hetero.
Christianity is based upon duality and not triality or trinity!
Mind (the internal soul, syntropy) is dual to matter (the external soul, entropy) -- Descartes.
According to Descartes the mind/soul is dual -- self duality.
Duality: two sides of the same coin (heads is dual to tails).
Optimism is dual to pessimism.
The optimist says the glass of water is half full, the pessimist says the glass of water is half empty -- two dual perspectives of the same thing.