Is this one connected curve, or two? Bet you can't explain why...

One of the most fundamental properties of geometry is connectedness: when a shape is a single continuous entity. But how do you define this idea precisely so that you can apply it even to extremely bizarre shapes in very strange spaces?

=Chapters=
0:00 - The Topologist's Sine Curve
1:54 - The goal of this video
3:06 - Path-Connectedness
6:56 - A new definition?
7:37 - Topology basics
10:27 - A snag
12:04 - Connectedness 2.0 and the Topologist's Sine Curve
15:48 - Who's right?
17:38 - The Ordered Square
19:32 - The connection between the definitions
21:18 - The underappreciated art of crafting definitions
22:16 - Brilliant ad

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CREDITS

The songs used in this video are (in order of first appearance) "Dream Escape", "Checkmate", "Rubix Cube", and "Twinkle in the Night".

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Thank you for your support!

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The animations in this video were mostly made with a homemade Python library called "Morpho". If you want to play with it, you can find it here:

[morphocular]

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!

[officially_certified_nerd]

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

[jenbanim]

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

[DebashishGhoshOfficial]

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

[daniloelias]

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!

[michel_dutch]

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

[MasterHigure]

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 5:30 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 8:25, that could've been a good fit here too.

[zapazap]

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

[SuperCornstock]

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!

[brucea]

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.

[bigbidnessborsalino]

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.

[dazperson]

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.

[williammanning]

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.

[dyld]

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.

[josvanderspek]

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!

[Boringpenguin]

21:12 A follow up video on this equivalence would be awesome.

[theultimatereductionist]

15:57 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.

[blacklistnr]

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.

[silvory]

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 2:30am 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

[abekolko]

My attempt at coming up with a definition for "connectedness, " as you mention at 2:30. 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!