They end up being the same, but y -x is more intuitive, and that's the convention to always subtract the initial or center point x from the final or displaced point y, because that's the value in 1D when y is displaced from x in the positive direction.
I understand intuitively why you {2,1} = {1,2} but when you take the cartesian product you mentioned how one gets ordered pairs the first slot comes from the set and the second slot from the second set so is there not some implication of structure there?
So yes there is more structure! Since you now have two sets rather than one you have to be careful which element comes from which set, and this is reflected by the ordering of the elements in the pair - hence additional structure than just a bare set by itself. It is still true that {2,1} = {1,2}, so the elements of one particular set in the pair could be interchanged and this would just produce one of the other possible pairs.
So in the last part, both 2 formulas are described in the topological space? I suppose the lowercase u is partial manifold as you will show in latter videos. Thus, the R^2 in this video is actually not from the coordinate in video manifold #3 (i.e. Cartesian coordinate, I suppose that is in Euclidean space). Am I right? Thanks.
Three remaining questions from the previous lecture. 1.you draw the S1 line(which is the real number "angle" line of the orange circle (from 0 to pi) and specifying two end points (which are purples)and you draw this end points in purple because you want to indicate you touch the orange loop( 0 to 2 pi) into two ends points of purple lines,right? Therefore the ends of the orange lines are purple? what i understand is : firstly you created the first orange loop and then create your second purple loop.Then according to the topological in-variance,you "map" this first orange loop into real number interval (0 to pi),then you let the end points of the orange loop to become equal which is the boundary condition (the reason you specify these two points to become purple is because you want to say you touch the orange S1 lines into purple lines but not S1 itself is purple therefore you specify these two end points as purple?) Then just like cross product,you can multiply each elements of the orange lines into the purple line and you can get the whole square. So my question is why do you draw the end points with different colours than the colour of the line interval 2.The shorter one (orange line) is clear but the longer line is not very clear. since the radius for each purple loops are differents if you put this loop into the outer part of torus or inner part of torus,where do you actually put your loop 3.in the previous lecture I observed that two purple lines have the same length but this cannot be true if you make a torus because one of the purple lines must be longer than the other purple lines(which forms the outer part of the torus?) For example if you want to make a annulus you need to use bend a longer purple lines which includes the shorter purple loop inside, therefore what do you actually do for this purple loop.
@@garytzehaylau9432 So I used two separate colours just to make it really clear which circle is which. It doesn't matter whcih circle is which since they are treated equally but it's helpful to keep track of how T^2 is formed out of two circles. I'm not quite sure what you mean with 1) so feel free to ask again if I don't really answer well enough. But if the colours are confusing you let me just explain abit more: The coloured edges represent the identified edge points of each circle, with the purple (vertical) edges being the identified end points of the orange line segments and orange edges the end points of the purple line segments. Each coloured line then represents the entire set of all pairs, (endpoint of one circle (eg purple point), entire other circle (eg orange line). The coloured lines are bounded by a point with an opposite colour, since each circle (coloured line segment) is attached to a single point on a different circle, which can be taken to be the boundary point of the attached circle. If the corners are confusing you due to the colours - really the corner points of the square should be a different colour to signify that they are all in fact the same point - the point where the two circles 'cross' (diagram I had on the right side of the board). Regarding 2 and 3) I think you are getting confused due to preconceived notions of distance which you need to be really careful with when dealing with topology. Because we haven't defined distances yet, when we draw these figures we have to realise that their geometrical representation in the figure does not reflect actual properties of the space. The annulus for example, the inner and outer boundary circles are drawn at different sizes - but remember since all circles are isomorphic, we encounter no difficulty in identifying the points of these seemingly different (but only in size!) circles. The best way to think about this is that the figures we draw are made of ideal rubber that we can stretch and deform (in a continuous way only) as much as we like and everything remains topologically equivalent! Specific to 2) Whether the purple circle forms the 'outer' or 'inner' edge of the equatorial part of the torus is also arbitrary, and can even be taken to be the circle that passes through the centre of the ring part as I drew on the right side of the board (purple circle passing through centre of orange circle) without loss of generality. Since really the cartesian product is telling us that a circle is attached at every point of the orange circle, the purple circle we are considering is only one of these such circles so we can really place it anywhere on the orange circle - the entire surface is then formed from an infinite number of these circles (one for every point on the other circle) filling out the entire torus (or fundamental region)
Now what I understood from this video that open set essentially means continuity. I think this is done in function spaces with the help of Cauchy sequence.
These videos are really nice.
You seem to prefer (y-x)^2 over (x-y)^2 in the video, but aren't the two expressions effectively the same? Am I missing something?
They end up being the same, but y -x is more intuitive, and that's the convention to always subtract the initial or center point x from the final or displaced point y, because that's the value in 1D when y is displaced from x in the positive direction.
Thank's for this, I'll probably follow the whole series. Keep it up!
I understand intuitively why you {2,1} = {1,2} but when you take the cartesian product you mentioned how one gets ordered pairs the first slot comes from the set and the second slot from the second set so is there not some implication of structure there?
So yes there is more structure! Since you now have two sets rather than one you have to be careful which element comes from which set, and this is reflected by the ordering of the elements in the pair - hence additional structure than just a bare set by itself.
It is still true that {2,1} = {1,2}, so the elements of one particular set in the pair could be interchanged and this would just produce one of the other possible pairs.
So in the last part, both 2 formulas are described in the topological space? I suppose the lowercase u is partial manifold as you will show in latter videos. Thus, the R^2 in this video is actually not from the coordinate in video manifold #3 (i.e. Cartesian coordinate, I suppose that is in Euclidean space). Am I right? Thanks.
Thank for sharing this video.
i still have questions in previous lecture but not this one.
Three remaining questions from the previous lecture.
1.you draw the S1 line(which is the real number "angle" line of the orange circle (from 0 to pi) and specifying two end points (which are purples)and you draw this end points in purple because you want to indicate you touch the orange loop( 0 to 2 pi) into two ends points of purple lines,right? Therefore the ends of the orange lines are purple?
what i understand is : firstly you created the first orange loop and then create your second purple loop.Then according to the topological in-variance,you "map" this first orange loop into real number interval (0 to pi),then you let the end points of the orange loop to become equal which is the boundary condition (the reason you specify these two points to become purple is because you want to say you touch the orange S1 lines into purple lines but not S1 itself is purple therefore you specify these two end points as purple?) Then just like cross product,you can multiply each elements of the orange lines into the purple line and you can get the whole square.
So my question is why do you draw the end points with different colours than the colour of the line interval
2.The shorter one (orange line) is clear but the longer line is not very clear.
since the radius for each purple loops are differents if you put this loop into the outer part of torus or inner part of torus,where do you actually put your loop
3.in the previous lecture I observed that two purple lines have the same length but this cannot be true if you make a torus because one of the purple lines must be longer than the other purple lines(which forms the outer part of the torus?)
For example if you want to make a annulus you need to use bend a longer purple lines which includes the shorter purple loop inside, therefore what do you actually do for this purple loop.
@@garytzehaylau9432 So I used two separate colours just to make it really clear which circle is which. It doesn't matter whcih circle is which since they are treated equally but it's helpful to keep track of how T^2 is formed out of two circles.
I'm not quite sure what you mean with 1) so feel free to ask again if I don't really answer well enough. But if the colours are confusing you let me just explain abit more: The coloured edges represent the identified edge points of each circle, with the purple (vertical) edges being the identified end points of the orange line segments and orange edges the end points of the purple line segments. Each coloured line then represents the entire set of all pairs, (endpoint of one circle (eg purple point), entire other circle (eg orange line). The coloured lines are bounded by a point with an opposite colour, since each circle (coloured line segment) is attached to a single point on a different circle, which can be taken to be the boundary point of the attached circle. If the corners are confusing you due to the colours - really the corner points of the square should be a different colour to signify that they are all in fact the same point - the point where the two circles 'cross' (diagram I had on the right side of the board).
Regarding 2 and 3) I think you are getting confused due to preconceived notions of distance which you need to be really careful with when dealing with topology. Because we haven't defined distances yet, when we draw these figures we have to realise that their geometrical representation in the figure does not reflect actual properties of the space. The annulus for example, the inner and outer boundary circles are drawn at different sizes - but remember since all circles are isomorphic, we encounter no difficulty in identifying the points of these seemingly different (but only in size!) circles. The best way to think about this is that the figures we draw are made of ideal rubber that we can stretch and deform (in a continuous way only) as much as we like and everything remains topologically equivalent!
Specific to 2) Whether the purple circle forms the 'outer' or 'inner' edge of the equatorial part of the torus is also arbitrary, and can even be taken to be the circle that passes through the centre of the ring part as I drew on the right side of the board (purple circle passing through centre of orange circle) without loss of generality. Since really the cartesian product is telling us that a circle is attached at every point of the orange circle, the purple circle we are considering is only one of these such circles so we can really place it anywhere on the orange circle - the entire surface is then formed from an infinite number of these circles (one for every point on the other circle) filling out the entire torus (or fundamental region)
@@WHYBmaths totally understand
very clear.
Shouldn't the discrete topology be 'tau = P(M)'?
I also have a same question. Should it be P(M) U ∅, or simply P(M)?
Sir!
Please upload more and more videos.
Thank you.
Now what I understood from this video that open set essentially means continuity.
I think this is done in function spaces with the help of Cauchy sequence.
i miss the dog
At least we have it drawn on the board.