at @4:10, the lecturer says that the pushout is the largest group with some property. Isn't it the smallest? Roughly speaking, the universal property of a pushout guarantees it is an initial object. So among all groups with such a property, 'roughly speaking' it is 'minimal/smallest' with such a property.... isn't it?
Does the intersection actually have to be open ? I thought that pathconnectdness and non-emptiness is sufficient. After all, when studying a very simple application of Van - Kampen, namely the wedge of two circles pi_1(S^1 ∧ S^1) isomorphic to Z * Z, the intersection of the two circles is simply a point, certainly not open.
Recall that the Van Kampen's theorem has as an assumption to consider an open cover given, let say, by A and B. Then, directly from the axioms of topological spaces, the intersection much be open. You couldn't have taken a open cover in the "counterexample" you're giving for the same reason. Possibly, you've taken A to be one of the circles and B to be the other circle. However, observe that neither A nor B are open in the induced topology of your space since the "union" point is not interior neither in A nor B. Therefore, you must take as small neighborhood of the "union" point both in A and B which gives you a small open section of the other circle. Therefore, the intersection is not a point but a small neighborhood of the union point.
1:00:23 Is it single A_a or at least single A_a. I assume partitions are obtained using Lebesgue number lemma so it shall be: at least single A_a (For example some point in intersection of A_a and some A_b)
WHY do all the open sets have a common point? Is the statement of van kampen that it's the free product of the the fundamental groups of the A_\alpha which contain the point? Since they are path connected their fundamental group is well defined without specifying a point, and this sure doesn't seem equivalent to that formulation. I don't understand how we are making all these paths to x_0, why would there be a common point? These things are meant to be a cover.
@@MWTan-ho2to A year late but I believe the point is we need the point in every set A_\alpha as we can't a priori assume the intersection of sets A_\alpha \intersect A_\beta is path connected (even though the individual sets are) as that is a hypothesis of the theorem
Both the sets have to contain the basepoint for us to justify each based loop having a finite factorization of based loops entirely contained in one set or the other.
Homotopic equivalence = duality! Union is dual to intersection. Null homotopic implies contraction to a point, non-null homotopic requires a second point (forming a line). Points are dual to lines, the principle of duality in geometry. "Always two there are" -- Yoda. Continuous (topology) is dual to discrete (quantum). Gravitation is equivalent or dual to acceleration -- Einstein's happiest thought, the principle of equivalence (duality). van Kampen's theorem requires equivalence or duality to be conserved at all times! The conservation of duality will be known as the 5th law of thermodynamics, energy is duality, duality is energy -- Generalized duality. Duality creates reality!
I got it. Really rather well explained. But I probably could'nt do the homework.
at @4:10, the lecturer says that the pushout is the largest group with some property. Isn't it the smallest? Roughly speaking, the universal property of a pushout guarantees it is an initial object. So among all groups with such a property, 'roughly speaking' it is 'minimal/smallest' with such a property.... isn't it?
yes it's the smallest
just a year late
Does the intersection actually have to be open ? I thought that pathconnectdness and non-emptiness is sufficient. After all, when studying a very simple application of Van - Kampen, namely the wedge of two circles pi_1(S^1 ∧ S^1) isomorphic to Z * Z, the intersection of the two circles is simply a point, certainly not open.
Recall that the Van Kampen's theorem has as an assumption to consider an open cover given, let say, by A and B. Then, directly from the axioms of topological spaces, the intersection much be open.
You couldn't have taken a open cover in the "counterexample" you're giving for the same reason. Possibly, you've taken A to be one of the circles and B to be the other circle. However, observe that neither A nor B are open in the induced topology of your space since the "union" point is not interior neither in A nor B. Therefore, you must take as small neighborhood of the "union" point both in A and B which gives you a small open section of the other circle. Therefore, the intersection is not a point but a small neighborhood of the union point.
1:00:23 Is it single A_a or at least single A_a. I assume partitions are obtained using Lebesgue number lemma so it shall be: at least single A_a (For example some point in intersection of A_a and some A_b)
I don't really understand why step 2 doesn't change the class of the word in the quotent group. Please, can anybody explain it to me ?
Ok, I got it. But here is another questuion. 1:04:53 maybe I am stupid. I don't understand couse of equivalence of this factorizations.
WHY do all the open sets have a common point?
Is the statement of van kampen that it's the free product of the the fundamental groups of the A_\alpha which contain the point? Since they are path connected their fundamental group is well defined without specifying a point, and this sure doesn't seem equivalent to that formulation.
I don't understand how we are making all these paths to x_0, why would there be a common point? These things are meant to be a cover.
the proof doesn't seem to depend on them having a common point
it doesn't have to be the same point every time
okay I was wrong, we do use a common point, but it's free so no problem
@@MWTan-ho2to A year late but I believe the point is we need the point in every set A_\alpha as we can't a priori assume the intersection of sets A_\alpha \intersect A_\beta is path connected (even though the individual sets are) as that is a hypothesis of the theorem
Both the sets have to contain the basepoint for us to justify each based loop having a finite factorization of based loops entirely contained in one set or the other.
1:03:20
Homotopic equivalence = duality!
Union is dual to intersection.
Null homotopic implies contraction to a point, non-null homotopic requires a second point (forming a line).
Points are dual to lines, the principle of duality in geometry.
"Always two there are" -- Yoda.
Continuous (topology) is dual to discrete (quantum).
Gravitation is equivalent or dual to acceleration -- Einstein's happiest thought, the principle of equivalence (duality).
van Kampen's theorem requires equivalence or duality to be conserved at all times!
The conservation of duality will be known as the 5th law of thermodynamics, energy is duality, duality is energy -- Generalized duality.
Duality creates reality!