Hi all! I wanted to quickly address a misconception that I've been seeing in a lot of comments. A number of you seem to be confused as to why a path is defined using a function, since that would seemingly exclude paths such as circles which fail the vertical line test. The thing to realize is a "path" here is not defined simply as a function that takes a real number input and then outputs another real number, but it outputs points in the space, which for a 2D space means entire (x,y) pairs. So a path in 2D space is a function that takes real number inputs in the range [0,1] and outputs (x,y) pairs, not just y-values. So for example, a full circular path can be defined as f(t) = (cos t, sin t) which describes a point moving counter-clockwise around a circle of radius 1 (if t ranges from 0 to 2pi). If you think of t as representing time, then this literally describes a point moving around a circle with speed 1. I hope that helps!

my one semester of analysis has finally paid off in the form of understanding this youtube video and knowing the definition of connectedness

Sin(1/x) is the maths equivalent of a black hole. It's just freaking awesome.

The structure of this video is fantastic! Every time I thought something was confusing or needed an example, it was immediately followed by further explanation or an example. Anticipating how people are going to follow along is difficult, so I'm very impressed with how well this video manages to do that

Exactly!! We need to know the motivation in arriving at a particular definition in the textbooks. This would make learning more interesting.

The topologist's sine curve is a very cool topological space, and it is a very natural place to get into the nitty-gritty of what connectedness means in all its forms. I think you have done that very well here. However, one point of criticism: I think it's wrong to entirely fail to mention around <a href="#" class="seekto" data-time="330">5:30</a> the possibility that there may be *more* than one value t that gives f(t) = (0, y). It doesn't change the result, but it does slightly complicate the argument (you need to choose your t* carefully), so I can understand glossing over it. But not mentioning it at all is far beyond just glossing over it, and I personally think that falls into the territory of "incorrect". You have a note for the pedantic when talking about neighbourhoods at <a href="#" class="seekto" data-time="505">8:25</a>, that could've been a good fit here too.

I am finishing my masters in physics and I must say this is the clearest explanation on this subject I've ever seen. Keep up!

Next topic: compactness. You can present different notions of compactness like you presented different notions of connectedness.

eyeopening and excellently narrated. The visuals and transitions were just perfect as well. Perfect video on the subject, thank you for the hard work it took to make this!

Absolutely fantastic video. Even though I encountered this example 5 years ago, I can honestly say this is the first time I actually truly capture the subtle difference between the two definitions. Also, kudos on the definition-motivation plea at the end. Great work!

After watching the intro, here's my guess: to connect the two "ends" of the curve around 0, those individual ends must have well-defined limits. Even though the overall curve does not have a well-defined limit at 0, you could imagine each individual end having one. However, for this curve, each side of the curve near 0 is just as badly-behaved as 0 itself.

I love how in the last moments of the video he showed a graphic of the epsilon delta definition of continuity that instantly made me understand it despite eluding me for years at this point. Excellent video if anyone's reading this.

Such a good watch! Very interesting topic and a well made video. Cheers!

<a href="#" class="seekto" data-time="1272">21:12</a> A follow up video on this equivalence would be awesome.

Interesting video. And I can certainly appreciate the point you made at the end about the importance of definitions and the difficulties mathematicians sometimes have formulating good definitions. I recall from my studies of calculus that it took a while for mathematicians to derive the definitions of limit and derivative that we take for granted today. Before our current definition of limit was proposed nobody could define limits in terms that did not verge on the mystic.

This is a lovely video. It took my right back to my first topology course. Thanks for sharing.

One of the best videos on connectedness i've seen so far, and i've seen a lot

Great video! I especially like the bit at the end when you compare semantics. I think math's greatest problem is that definitions usually throw away all context and motivation which makes them happen in the first place. It's like trying to describe an algorithm in assembly/binary, yeah it's useful for execution, but beyond terrible for understanding the ideas at play.

Brilliant concept, wonderful video. Great job.

Very well-made!!! Thank your for your beautiful piece of explanation!! 😻😻

Fantastic !! as usual for your standard !!

I love point-set topology for how completely insane and seemingly contradictory it is despite being entirely consistent. You didn't even get into locally connected and how it's separated from connected. Also, I don't think the textbook definition of connected is that hard to explain, since a closed set is defined to be the complement of an open set. So if a non-trivial set in the space is both open and closed, then it's a non-empty open set, whose complement is also non-empty and open.

Loved this video! I managed to guess the path definition but the subset definition was really cool.

this was such a beautiful video i HAD to leave a comment, like, and subscribe. what i've found really beautiful in mathematics, is when we start coming up with definitions to seemingly answer an extreme case of an intuitive problem, the solution is just "whatever you so desire". it took me sometime to appreciate this beauty, because at first it feels like a slap in the face, like when you're in an adventure story to find the truth of the world, meets an all-knowing-being find what is the meaning of life and they ask back at you "what is your meaning of life?". and this video really encapsulates that feeling. also that 1st textbook definition of "connected" really blew me away. it's almost like they used the aforementioned (two open disks touching at their boundary, a.k.a. a union of two non-empty disjoint open sets) to come up with a definition. fascinating.

So nice to see a video on point set topology. The _best_ topology. :)

"Just as every theorem needs a proof, every definition needs a motivation" It feels like I'm the only one in my math class thinking this, including the teacher ;)

Awesome explanation! keep going!! looking forward for more such amazing Math Content

These are extremely excellent videos. Hope you do way more especially about the weirdest math topics! Thank you

I subscribed immediately 'cause of this awesome vid! As a physics student, it's been difficult for me to have an intuitive grasp of topological properties like this. I hope you do a video about COMPACTNESS!!!

Another question: instead of the topologist's sine curve, if we modify it such that we only add the point (0, 0) instead of a whole strip from (0, -1) to (0, 1), would the resulting set still be connected?

<a href="#" class="seekto" data-time="957">15:57</a> My mind went immediately to the Touching Subsets definition of connectedness and concluded the Topologists' Sine Curve is connected, before you reminded me about the path-connected definition.

I totally agree with you, each definition should be motivated before diving into the proof

You are one of the only math creators I can understand without getting bored

I have been studying civil engineering since last October and am struggling to grasp the concept of continuous functions. Thanks to your videos, some things have started to click and the gears are starting to move. That may be as well because I have been studying for the upcoming math exams for the entirety of last week, but hey, i’ll take what I can get.

thanks for making the end of this video, you didn't have to include that boundary, but you did ❤

What a surprisingly cleanly-explained video

This was actually a really good video wow, never thought I'd waych it for 20minutes, thank you s lot ^^

Excellent explanation loved it ❤️

I'm taking topology this semester, so seeing this video made me very happy :)

My attempt at coming up with a definition for "connectedness," as you mention at <a href="#" class="seekto" data-time="150">2:30</a>. I wanted to see how I'd do before I see the answer. Define a Connector set as being constructed by the following process. Start with a well ordered set of m points with m>1. Call this the seed set A = {a_i| i from 1 to m inclusive}. The connector set is the union of all of the line segments connecting a_i to a_i+1 for i ranging from 1 to m-1. Now suppose we want to assess if a region R is connected. The set R is connected iff for all r_1 and r_2 in R, there exists a connector set A containing r_1 and r_2 st A is contained within R. Edit: Ah I now see this definition only works if R is of at least 2 dimensions everywhere. Tricky!

It's simple. We have a graph defined as f(x)=sin(1/x); at x=0 it has no value because decision by 0 is not possible. Therefore the curve is disconnected at a single vertex, exactly at x=0.

Awesome video. Thank you

This is just fascinating and mysterious, and maddening to me, so entertaining, beats all tv programs, tbh

Fascinating!

Just a truly great video!

This channel needs a few million subs, So good

This was a really well made video

Incredible video! ❤

I'd say they don't agree because the topological definition doesn't include the requirement of a limit, only that the sets share a boundary which is in the whole set, which I interpret to mean the function has (at least one) y value for each x value. Even in the 'path definition', if you remove the requirement that the limit exists, you can define a (discontinuous) function that works (but again that is not the definition of "path")

Masterpiece of a video

It will be amazing it you do a similar video about compactness

fascinating video, when i first saw the proposed question i immediately thought about continuity and came to the conclusion that the curve is not connected, as the limit of the function at x=0 is undefined therefore the curve is not continuous at x=0. but the rest of the video really intrigued me as it gave me a new perspective on the idea of "connected"

This sine function is used in Physics in what is called "asymptotic methods" usually in integrals and usually disguised as a complex exponential.

Very well-made video!!

As a hint for getting to the third definition I came up with an intermediate definition that is still somewhat intuitive. Instead of defining a boundary point you simply define a limit point of a set A as a point x ( either inside or outside A) in which every neighborhood of x contains some member of A. Then the closure of A is just A plus all of A's limit points (which, by definition, will include all the boundary points of A as well as interior points). Then the equivalent definition of "touching" between two sets A and B will be that either 1.) A contains a point in the closure of B, or 2.) B contains a point in the closure of A. The proof that this definition of "touching" is equivalent to the boundary-based definition shouldn't be too hard. This is the only missing link I have yet to think through. Then the next second step is to refine the definition of touching for the case where we just assume nothing exists outside of A and B. This is a little cleaner because limit points outside of A U B no longer exist. They are now outside of the domain, i.e. outside the universe we are considering. They become topologically analogous to infinite limits (kind of an aside, but interesting to think about). In any case, if we throw away all the limit points external to A U B, then the definition of "touching" reduces to either A or B having ANY external limit point ( since if a point is not in A, it must be in B, and vice verse ). At this point the only thing left to do is show that "both open and closed" is equivalent to "contains all of its limit points" within the new restricted universe where nothing exists outside A U B. Once you do that you have the third definition. I know I'm close because "contains all of its limit points" is a pretty popular alternate definition of closed in real analysis texts. This would be so much easier to show using illustrations but I don't have the talent of the author of this video unfortunately. :(

Very nice and clear as usual. Just to nitpick, the sentence _"the sets contain at least part of their common boundary"_ *implies* that two sets touch, but it's not a *definition* since two sets can touch without having a common boundary, e.g. the sets of points at a distance of at most 1 and 2 respectively from the origin.

There's always the "room" for rationals (b/c compact rows), and more that infinitely much space for real numbers. The boundaries got duplicated at <a href="#" class="seekto" data-time="604">10:04</a>. Before that, they we correctly shown belonging to only one set on cutting.

The definition I came up for "Connectedness" was: Any two shapes of bodies that share, at any point in their area/volume, the same spatial coordinate/s, as infinitesimally small said coordinate is, as long as both shapes/bodies share this coordinate, they're connected. The moment any two shapes/bodies stop sharing their coordinates, they're disconnected. I know it's kinda weird talking about connectedness with plural examples, but it is that way so it can be as general as it can be whilst also being as precise as I could make it.

Topologist's sine curve is one of my favourite so I watch this! If I don't misunderstand, the definition of the curve T you used is T:=the union of left part L, right part R and Y={(0,y): -1

Your animations look good, helpful too.🙂

Oh yeah that is good Stuff man. thanks, i hope there is more. Keep up the good work!!thanks

I would argue that the curve with the line along the y-axis is still connected with the path definition (Disclaimer: I am not a mathematician, so feel free to correct me if I'm misusing terminology; more notes at bottom). There are only two possible scenarios for the "endpoints" of the separated curves without the vertical line (these "endpoints" reflect the y-value of the function immediately to the left and immediately to the right of x=0). The endpoints are either located at effectively the same point to create what would normally be a removable discontinuity or located at different y-values to create what would normally be a jump discontinuity. These types of discontinuity usually refer to functions where the limits at the x-value are known. In this case, the limits from the left and right are undefined due to the oscillating pattern. However, the function is bounded between two y-values, so we know that the "endpoints" of both sides will each have a y-value between -1 and 1. This would mean that adding a line at x=0 that ranges from y=-1 to y=1 should attach to both disconnected "endpoints" regardless of where they are positioned relative to the y-axis. Although we cannot determine the precise values of the path function as it meets the y-axis, similar to the way that we cannot determine a finite limit at x=0, an indefinite path function must exist that traces the vertical line to connect the two segments. Note: This is the result of a bunch of concepts and theorems back from Calculus 1 that have been scrambled together from memory to try to make something that resembles a decent explanation. The main inspiring concepts aside from discontinuity were the Intermediate Value Theorem and the Squeeze Theorem. It's also <a href="#" class="seekto" data-time="150">2:30</a>am as I'm finishing this (I got hyperfixated on number stuff again), so please let me know if there's something that doesn't make sense, and I'll try my best to clarify later

Before I watch the rest of the video, you asked for my thoughts. I think the graph... is both connected and disconnected. On one hand, in the same way that there are infinitely many values between 0 and 1, now we find that no matter how far we zoom in on the y-axis, we can always zoom in further and insert a new point on either side that will have a different y-value. There is no meaningful "final" value of y for the "final" position x immediately below and above 0, so there is no well defined "attachment point" to which the y-axis can be "connected" with the curves on either side, hence disconnected. But, this oscillation continues infinitely, so there is no break or discontinuity between the y-axis and the curves, hence connected. Can limits help with this, maybe? As they say on "Hollywoo Stars and Celebrities: What do they know, do they know stuff? Let's find out," let's find out!

I would argue that the path definition actually leads to Connected. The problem with the argument in the video was that it assumed we could only include 1 point in the center line on the path, while we can actually include multiple. A parametric function, as used here, is able to have different outputs with the same x-coordinate.

There's a fun puzzle related to this. Consider a (closed) square ABCD, with the corners labeled so that A is opposite C and B is opposite D. Find two disjoint subsets of the square, one containing A and C and one containing B and D, such that each subset is _connected_ (in the touching subsets sense). Hint: both subsets must be neither open nor closed.

Thanks for the video. I just want to know that what is the name of the song in <a href="#" class="seekto" data-time="228">3:48</a>. Thank you so much

It's a slightly more advanced topic that I never fully got through, but another really interesting topic involving definitional motivation is the concept of dimensionality. Hausdorff dimension vs. topological dimension would have a discourse roughly analogous to path-wise connectedness vs topological connectedness. Hausdorff dimension is more complicated, but also more intuitive IMO. In the case of Hausdorff dimension, instead of the topologists sine curve, pathological cases where intuition fails are fractals! You'd think that the boundary of a 2 dimension object must be a 1 dimensional object, but the boundary of the Mandelbrot Set has Hausdorff dimension 2. In other cases you can get a fractional Hausdorff dimension ( hence the name "fractal" ).

The "textbook definition 1" is pretty clearly equivalent to the one you explained imo.

Very nice video. I hope you'll soon get a lot more subscribers.

<a href="#" class="seekto" data-time="1278">21:18</a> “But in any case, one of the things I hope you take away from this video is an appreciation of the ingenuity that goes into crafting a good mathematical definition. So often in Math, the focus is on the big theorems, proofs, and calculations, but I feel we don’t often pause to appreciate all the creativity and cleverness that go into some of the definitions. Maybe it’s partly because textbooks and lectures often just state definitions unmotivated before quickly moving on to the theorems and proofs. *But just as every theorem needs a proof, I think every definition needs a motivation.* And it’s an important mathematical skill to cultivate too! One that I worry too often goed unexercised. So the next time you encounter a strange mathematical definition out in the wild, take a minute to flex that mathematical muscle and try to understand what motivated it, and also appreciate the work that went into crafting it.”

Really nice introduction to these concepts! However, an important thing I always try to teach my students is that topological notions are always relative. An Intervall in R may be open, however the same interrval in R^2 will not be w.r.t. the standard topology on R^2. Usually the topologist's sine curve is not just considered as a subset of R^2 with its standard topology, but rather as a topological space in it's own right, by giving it the subspace topology. In this case your argument, that all points of the curve are boundary points, is incorrect. That's also why connectedness of a subset of a topological space is defined with open subsets of the subspace topology, not with open sets of the bigger space.

Wonderful video!

<a href="#" class="seekto" data-time="972">16:12</a> Holy shit! That philosophy hit me deeper than it should

Excellent video!

It’s two unconnected curves. The sine function is odd, -sin(x)=sin(-x) LH and RH values have opposite signs. True even as x-> infinity. Conversely, the cosine is even, cos(x)=cos(-x) LH and RH values are always equal, and “connect” as x-> infinity.

In my head, I define connected shapes as shapes that, at all points, have a continuous link. For line and curves, each point must connect with exactly two points to count. More and that's seen as passing through itself at at least one point. For geometric shapes you define an inside and outside. Outside is defined as having access to infinity without touching inside. In the case of "hollow" shapes like a ring, inner "outsides" are considered their own shape and these follow their own rules for defining in/out, and their inside is considered outside for the larger shape. If both insides can continuously connect without breaching a border or the outside, they are connected. So if two points are seperated by the border of a shape, they cannot connect. By my definitions, this sine function is connected as long as no points from either half touch x=0 aside from the final "infinity" points of each wave. I should mention that I am, in fact, a gay dog on the internet with no qualifications in this field, so my definitions may be incomplete or flawed. This is just an intuition opinion born of boredom

It's not clear to me from the description of the ordered square in this video what its topology is. Perhaps a total order in a set defines a topology by taking all the intervals (a,b) to be open? I can probably fill in the gaps myself, but I think I need more detail to understand this example of connected space that is not arc-connected.

Very interesting video. But please kindly clarify regarding the Ordered Square (<a href="#" class="seekto" data-time="1061">17:41</a>) 1. Why a space-filling curve, such as the (infinite iteration of the) Hilbert curve cannot be considered an ordered square? 2. If it is indeed possible to use the Hilbert curve as a 1D path through the ordered square, then would this also enable us to use the path idea to define connectedness? Thanks in advance to anyone explaining these questions.

Before seeing the definition, my definition of connected for a graph is that for any point there's another point arbitrarily close to it from both sides of it and my definition for connected in topology is that for any two points there's at least one conneted (this meaning of connected would be the definition from the graph) path that starts at the first point and ends at the second and doesn't have any points not in the set inside of it

I started liking your videos before I even watch them!

<a href="#" class="seekto" data-time="1092">18:12</a> "this picture is a lie". Love it!

The topologist' sine curve is defined as the closure of the set (x,sin(1/x)), 0

What about the function y=sin(1/(x^2)) ? The curve will approach zero with exactly the same y value from left and right, so both ends have to touch in the center, right?

awesome video as always

The function is clearly undefined at x = 0, but its behaviour infinitely close to zero is interesting. f(-𝛿) = -1 and f(𝛿) = 1 for 𝛿 -> 0, both sides approaching zero infinitely fast towards the origin. So in a sense, f(±x -> 0) maps to every value between -1 and 1 except 0.

I like to think of functions in terms of sets, so for me "connectivness" is the ability of sets to have common members (aka intersect)

Hi, the graph (sin(1/x)) is connected at zero because it is a normal graph just like (sin x). They are both normal graphs, I used Desmos to graph (sin(1/x). If we graph (sin x) and (sin(50x)) then we get 50 oscillations per unit of length than with the 1 oscillation with (sin x) .And we can continue up to (sin (infinity*x) which will get us an infinite number of oscillations per unit of length which is the same thing as (sin(1/x)). Thanks for your educational videos.

short answer: lim x->0 (x) => infinity => infinite oscillations remediable gap in definition at x = 0

Here's my try before I continue. If a shape is connected, you can take any arbitrary pair of points and draw a line between them that does not cross the boundaries of the shape. I do not just straight lines

I would say according to physics they are connected in a state that is similar to a particle in superposition, it's specific values is not 1 or zero like normal particles, but in fact both (and everything in between at the same time). However this property doesn't make the particle different, and it's not defined as one or another, just both. Assuming math can be stated to in some way describe the physical world, then therefore this wave could represent a particle in said superposition.

<a href="#" class="seekto" data-time="21">0:21</a> is that a good way to think about the reciprocal in all cases and if not, which ones?

@Morphocular regarding the addition of the points of the segment [-1,1] on the y axis to either the right side or the left side: I understand that when we add it, we end up having a connected set, but that set is not the set of points that are on the curve y=sin(1/x) (i.e. the graph of the curve). Again, the graph of sin(1/x) does _not_ contain the segment, so why are we adding points to it that don't belong to the set in the first place? As I understand you showed that an augmented set is connected but that doesn't mean that the original set was also connected. Could you elaborate on this please?

Buen trabajo, felicitaciones. Labor de alto valor social...

Am I correct that you could remove the center segment except for any one point on it and the set would still be connected?

Because sine is a cyclical function you can substitute 0 for 1/0 at the origin. Or another way of looking at it is if you take the mean value of any 'path' of length 0, the minimum and maximum (and therefore the mean) will be the same. The 0 length path at the origin has a different minimum and maximum but as it covers the full range of the functions values, they are by definition the positive and negative extremes for that function and the mean value of 0 is correct.

I played around with Desmos and input other trig functions of 1/x, with similarly brain breaking results. I also input tan(x!) and part of the right side of the graph looks like a solid colored rectangle and inputting csc(x!) and sec(x!) look like a similarly solid rectangle with a square cut out of it.

Great video! Doesn't the second definition require a metric? Given that a ball is defined as points within a certain distance of another point. How could we define "connectedness" without relying on metrics?

<a href="#" class="seekto" data-time="151">2:31</a> The first definition that comes to mind is: A shape S is connected iff all of its points are connected. Points x1 and x2 that belong to S are connected iff there is a point x3 that also belongs to S and distance(x1,x2) is greater than both distance(x1,x3) and distance(x3,x2). Not sure if I can define it without a distance function. And not sure if it's correct definition yet.

<a href="#" class="seekto" data-time="97">1:37</a> Connected, but NOT in the classical sort of math way. This connection is like what "imaginary-numbers" are to regular numbers, in this case that would be an "imaginary-connection". (or similar to what quantum entanglement is.) In what I call "classical math" it mean that a line is singular & continuous & infinitely thin. But in this connection you can think of the line as becoming WIDER, to the point that EVERY point within the oscillation is the same point on the same line, regardless of what Y-value you choose.

<a href="#" class="seekto" data-time="148">2:28</a> Definition 1: If you can draw an (infinite length?) line between the 2 shapes = disconnected. Definition 2: If you can draw a circle around one of the shapes, without encircling the other, then the shapes are disconnected. Definition 2b: If you follow the circumference of the first shape & does NOT end up on the second shape at any point during 1 revolution = disconnected.