[HN Gopher] This open problem taught me what topology is [video]
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This open problem taught me what topology is [video]
Author : surprisetalk
Score : 473 points
Date : 2024-12-25 06:08 UTC (1 days ago)
(HTM) web link (www.youtube.com)
(TXT) w3m dump (www.youtube.com)
| Oarch wrote:
| It's great that he revisited this problem.
|
| It was his original video on this topic which had me instantly
| hooked to 3B1B all those years ago.
| Darthy wrote:
| I've got a problem with that video that starts at 4:15. He seems
| to jump to the conclusion that for every midpoint there is only 1
| distance. But that midpoint is formed by picking 2 points on the
| edge, and one could easily pick two other points on the edge that
| have the same midpoint (but have a different distance). He did
| not address that point at that point in the video, and for the
| next 2 minutes I kept raising that point in my mind. After he
| continued down that path not addressing that point, I felt that I
| must have missed something, or that more intelligent math viewers
| would have solved that open question in the mind in seconds and I
| am not mathematically inclined enough to be the target audience.
| And I stopped watching that video.
|
| I think good educational videos are the result of a process where
| a trial audience raises such points and the video gets constantly
| refined, so that the end video is even good for people who
| question every point.
| Avshalom wrote:
| he maps two points (by using their midpoint) and a distance to
| the (x,y,foo) if it was two different points with the same
| midpoint but different distance it would map to (x,y,bar)
| suryajena wrote:
| Great point now we can raise the issue and he will do a
| revision 3, with even better explanation for those issues just
| like in the books.
| darthoctopus wrote:
| this is not a conclusion that he jumps to! all that is stated
| is that there is a mapping from every pair of points on a curve
| to a set of 3D coordinates specified by their midpoints and
| distances. there is no requirement for uniqueness here. in
| fact, the whole point of this is to turn the search for an
| inscribed rectangle into the search for two pairs of points on
| the curve that have the same midpoint and distance --- this is
| stated just 1 min 15 seconds after the timestamp that you point
| out.
| Masterjun wrote:
| He addresses this at 9:00 in the video. You're thinking of a
| function graph, but he never made a function. He just sets up a
| visualization of a set of 3D points.
| raincole wrote:
| > I think good educational videos are the result of a process
| where a trial audience raises such points and the video gets
| constantly refined, so that the end video is even good for
| people who question every point.
|
| It would be at least as long as a one-semester course in
| typical math major then.
|
| To address your specific question: he doesn't assume each
| midpoint has only one distance _at all_. He doesn 't say it and
| the visualization doesn't show it as so.
| mauricioc wrote:
| The function defined in the video is "Given a pair of points A
| and B on the curve, output (x, y, z), where (x, y) is the
| midpoint and z is the length of the segment connecting A and
| B", and the pictures are of its image, not its graph. But if
| you define it visually, then it's very natural to misunderstand
| it the way you did, since the picture looks a lot like a
| function graph of a function which takes midpoints (instead of
| pairs of points) and returns the distance corresponding to that
| midpoint (which is not well-defined, as you pointed out). If
| this happens, the viewer is then completely lost, since the
| rest of the video is dedicated to explaining that the domain of
| this function is a Mobius strip when you consider it to consist
| of _unordered_ pairs of points {A, B} (as one should).
|
| Ultimately, if you don't have a 100% formal version of a given
| statement, some people will find a interpretation different
| from the intended one (and this is independent of how clever
| the audience is!). I think 3Blue1Brown knows this and is
| experimenting with alternate formats; the video is also
| available as an interactive blog post
| (https://www.3blue1brown.com/lessons/inscribed-rect-v2) which
| explicitly defines the function as "f(A, B) = (x, y, z)" and
| explains what the variables are.
|
| The fact that "given a large enough audience (even of very
| smart people), there will be different interpretations of any
| given informal explanation" is a key challenge in teaching
| mathematics, since it is very unpredictable. In interactive
| contexts it is possible to interrupt a lecture and ask
| questions, but it still provides an incentive to focus on
| formalism, which can leave less time for explaining
| visualizations and intuition.
| looneysquash wrote:
| I'm don't feel like I really get the distinction between a
| mapping and a function, or a visualization and a graph.
|
| But he was careful to point out that it wasn't a graph.
|
| To me the key point is that the input is all three variables,
| the two points and their midpoint, and not just the midpoint.
| suryajena wrote:
| This video if it was a scientific paper I would have visualised
| absolutely nothing. I don't know that if we can submit/embed
| animations instead of PDFs for university classroom work/
| scientific papers, because that's really much better than having
| to read papers/PDFs that is so incomplete without the right
| imagination/visualization of the problem. The last time I was
| giving a mock seminar in my university using a GIF to explain the
| RRT algorithm I was warned to not use animations in presentations
| . . . I mean either it was really not that helpful to visualise
| the solution or it has to do something with age old standards
| that needs to be revised. I mean figures can only do 3 or 4
| frames isn't more frames better.
| szundi wrote:
| What if you print it? Your presentation is useless then /s
| yapyap wrote:
| 1 frame at a time
| formerly_proven wrote:
| PDF supports embedded, interactively manipulatable 2D and 3D
| graphics/objects.
| atoav wrote:
| That word "support" does a lot of heavy lifting there. A bit
| like in "Email supports end to end encryption".
|
| You are not wrong, but if you had to bet your life on
| somebody being able to get the information and you don't know
| _how_ they are going to view that PDF would you do it?
| ykonstant wrote:
| I've never bothered to look into a TeX based way to do this;
| is it something that can be done with TikZ/PGF?
| Qwertious wrote:
| PDF supports being printed out onto paper.
| dsign wrote:
| Careful where you say that :-)
|
| In 2002, when I was doing my second year at college, my
| professor was cool enough to let me submit an animation of the
| self-balancing insertion algorithm for AVL trees. Those were
| the years of Macromedia Flash and Director. It was a cool
| project, and I wish I had kept the files. Overall, it was a
| highly technical thing.
|
| Twenty and so years later, I still do animations, even if only
| as a hobby. These days I use Blender, Houdini, and my own
| Python scripts and node systems, and my purpose is purely
| artistic. Something that is as true today as it was twenty
| years ago is that computer animation remains highly technical.
| If an artist wants to animate some dude moving around, they
| will need to understand coordinate systems, rotations, directed
| acyclic graphs and things like that. Plus a big bunch of
| specific DCC concepts and idiosyncrasies. The trade is such
| that one may end up having to implement their own computational
| geometry algorithms. Those in turn require a good understanding
| of general data structures and algorithms, and of floating
| point math and when to upgrade it or ditch it and switch to
| exact fractions. Topology too becomes a tool for certain needs;
| for example, one may want to animate the surface of a lake and
| find out that a mapping from 3D to 2D and back is a very handy
| tool[^1].
|
| I daresay that creating a Word or even a Latex document with
| some (or a lot of) formulas remains easier. But if I were the
| director of a school and a student expressed that videos are
| easier to understand, I would use it as an excuse to force
| everybody to learn the computer animation craft.
|
| [^1]: Of course it's also possible to do animations by simply
| drawing everything by hand in two dimensions, but that requires
| its own set of skills and talent, and it is extremely labor-
| intensive. It's also possible to use AI, but getting AI to
| create a good, coherent and consistent animation is still an
| open problem.
| jltsiren wrote:
| If you need to visualize an algorithm in a talk, the usual
| approach is having a few slides representing the key steps
| instead of an actual animation. That way you can adapt the pace
| to the audience, stop to answer questions about any individual
| frame, and jump back to previous frames when necessary. People
| often find animations on slides distracting, and the forced
| pace is almost certainly wrong. And if the animation is longer
| than a few seconds, the talk stops being a talk and becomes an
| awkward video presentation instead.
| klysm wrote:
| > People often find animations on slides distracting, and the
| forced pace is almost certainly wrong.
|
| I completely disagree. Animations can be appropriate, but
| people have formed dogmatic generalizations due to shitty use
| of gifs
| leoc wrote:
| Eh, it's charming but honestly, for someone who doesn't already
| know the maths it's still just edutainment, as it leaps from
| trivial to incomprehensible in the blink of an eye.
| EdwardCoffin wrote:
| I don't think he's trying to make people understand the proof,
| rather to show them that topology really has an application for
| problems that aren't themselves topological in nature, and it
| is comprehensible enough for that purpose.
| throwaway98797 wrote:
| useful to smooth brains like me
|
| also meta lesson on how useful extra dimensions can be
| ykonstant wrote:
| This comment made me wonder if there is an analogous
| "inscribed cube" problem in three dimensions which is easier
| for smooth closed surfaces (>=V<=)
| mettamage wrote:
| I disagree, I'm not well-versed in math but I felt I could
| follow most of it.
|
| What I don't get though is the jump from the mobius strip to
| the klein bottle.
|
| He just goes and does it and duplicates the surface to reflect
| it to the original one. I do understand to some extent that
| once you have to assume the klein bottle is the shape you're
| looking for that because it's self intersecting, it must mean
| that you have 2 different points on that same surface and
| therefore 2 lines of equal length with the same midpoint.
| scrollaway wrote:
| The point of the jump is that if you want to track an extra
| coordinate and visualize it with the restrictions he
| mentions, then the klein bottle is the correct topology (the
| correct "visualization").
| mettamage wrote:
| Oh, so you mean to say that:
|
| 1. The positive surface is for tracking one midpoint for
| coordinates A and B
|
| 2. The negative surface is for tracking another midpoint
| for coordinates C and D
|
| Together it's a klein bottle. Klein bottle's always
| intersect, so therefore there's always an intersection of
| the two midpoints, which is why there's a set of points A,
| B, C and D such that line segments A and B are equally long
| as C and D going through the same midpoint.
| level3 wrote:
| The "positive" surface already contains all the necessary
| points. It's hard to prove that this surface on its own
| intersects with itself, but turning it into a Klein
| bottle makes the proof easy, since it's already known
| that the Klein bottle must intersect with itself when
| embedded in 3-D space.
|
| It takes some rigor to ensure that mirroring the surface
| and turning it into a Klein bottle doesn't introduce a
| problem that would invalidate the proof, but the idea is
| this:
|
| 1) The surface exists only in the "positive" area above
| the x-y plane, and the mirror exists only in the
| "negative" area below the x-y plane.
|
| 2) The two surfaces only share the points on the original
| curve (on the x-y plane), and these points correspond
| only to the trivial cases where A=B. The surface and its
| mirror don't intersect anywhere else.
|
| 3) The resulting combined surface is a Klein bottle in
| 3-D space, which must intersect somewhere. Because of 2),
| that intersection must either be in the positive space or
| the negative space. Either way, that means there is an
| intersection in the original surface.
|
| As briefly mentioned in the video, it's critical that the
| original constructed surface is only in the positive
| area, because otherwise when you mirror it and then turn
| it into a Klein bottle, the required intersection might
| just be the surface intersecting with the mirror, and not
| within the original surface itself.
| pfdietz wrote:
| The surface with the interior of the loop added forms
| something called a projective plane. A Klein bottle is just
| two projective planes glued together. Neither can be embedded
| in R^3 without intersections.
| sizzzzlerz wrote:
| In many ways, I agree. I have an engineer's understanding of
| math for my discipline but topology is most definitely not one
| of them. Through his graphics, I could most follow the gist of
| what he was attempting to get across but when it was over, I
| honestly had to ask my self, what did I just watch. Perhaps
| watching it again, really concentrating on it, and trying to
| understand, might help, but, in reality, it is so far out of my
| interest zone, I'll never do it.
| wholinator2 wrote:
| Well that's the "edu" part of edutainment. Sometimes you've
| gotta rewind or pause and think about what's being said to make
| sense of it. I do understand that sometimes videos go way too
| fast and leave tons of stuff out and that's very frustrating
| but 3b1b is a pillar of the community for very careful and
| complete descriptions of things. But then also the "tainment"
| part would signal that there's no need to watch if you're not
| interested.
|
| But all this could be my bias of having some math background,
| though never having studied topology or even analysis from
| anything like a class or textbook. Felt like the video was
| aimed directly at people like me
| CT4u8798 wrote:
| I have no clue about maths beyond extremely basic stuff, but am
| fascinated by this sort of thing, and I need pictures to
| understand stuff like this. What an excellent video. During it,
| when they introduced how you can map the 2D to 3 dimensions, my
| initial thought was "I wonder if this is how you could map 3D
| into the 4th dimension?". Then later they mentioned 4 dimensions.
| This is something I cannot visualise or really understand.
| chrsw wrote:
| I gave up on trying to visualize 4 dimensions. I don't know if
| it's possible. Instead I just try to think of 4D as more of
| ideas and less geometry: rules, consequences, capabilities,
| etc. We can do the same thing in 3 dimensions by saying things
| like "two objects can't exist at the same place and at the same
| time" or "parallel lines meet at infinity" or "parallel lines
| never meet" or something. We usually don't do that for 3
| dimensions because we have visualizations and intuitions which
| we can use instead of breaking everything down formally all the
| time.
| whatshisface wrote:
| Time is nature's forth dimension, so I think considering the
| various stages of a slice moving through a four dimensional
| object at once counts as a visualization.
| CT4u8798 wrote:
| Donnie Darko style.
| philipov wrote:
| Time is not a dimension of the same kind as spatial
| dimensions. It has a different metric and you can't move
| freely back and forth on it. When you rotate on the XT
| plane, it doesn't mean the same thing as rotating on the XY
| plane. It is not a good candidate for the sort of fourth
| dimension we're interested in.
| whatshisface wrote:
| The Euclidian group in four dimensions is _a_ fourth
| dimension, but the Lorentz /Poincare group is _the_
| fourth dimension. ;)
| gf000 wrote:
| My understanding is that time can _be_ a 4th dimension,
| but n-dimensional spaces themselves are simply a very
| basic mathematical structure, where a point can be
| described by n numbers (you can actually be abstract even
| in that, no need to stick to rational numbers, I
| believe).
|
| As long as you can map time to a number line, it's a
| valid representation. We just happen to have hardware
| acceleration for 3-dimensions, and the 4th is just
| completely unintuitive to us.
| BriggyDwiggs42 wrote:
| There's a video from the same channel on visualizing
| quaternions as a projection into 3d that was really fun for
| this. Only a restricted section of a 4d space, but i feel
| like the principle generalizes a little because of the idea
| of, like, imagining one 3d space thats finite as equivalent
| to an infinite 3d space, just stretched
| everydayDonut wrote:
| I've always wanted to make a 4d space in VR. That way it's
| only one dimension higher, technically. Could help to
| visualize it in a way that hasn't been done yet
| rafabulsing wrote:
| There's a 4D mini golf VR game you might be interested in
| checking out. It's called, uh, 4D Golf. Creative! I've not
| played it myself, but it's on my list. I hear it's pretty
| cool!
| klysm wrote:
| > I have no clue about maths beyond extremely basic stuff, but
| am fascinated by this sort of thing, and I need pictures to
| understand stuff like this.
|
| Fascination is all you need. I find many people have a lot of
| self-limiting beliefs around math. There's many reasons for
| them to develop, but I firmly believe that many people are
| legitimately interested in mathematics and have the capability
| despite their beliefs.
| chrsw wrote:
| One of the problems with math, like a lot of things, is that
| even though you may find it deeply interesting and
| fascinating and you may even see great utility in it,
| becoming an expert is very difficult and is fraught with a
| lot of failure which many people can't, or won't, stomach.
| gf000 wrote:
| I guess that's true for most things. Say, learning to play
| an instrument can be similarly difficult at first.
|
| Motivation is vaning, you need discipline to actually stick
| to something and get better at it. But even getting better
| day-by-day by only a tiny percentage will result in huge
| gains over long periods.
| bddg22 wrote:
| Haha great to see him mentioned: Lobb taught my Linear Algebra 1
| course a few (god im old) years ago. Excellent prof, and we still
| laugh over the looks of despair he gave us when we didn't get
| something.
| jebarker wrote:
| I loved this. I did my PhD in algebraic topology, but studied
| lots of topology so was familiar with this material. I doubt I
| could ever have explained these concepts so clearly or tied the
| esoteric world of topology to a "practical" problem.
|
| Since my PhD I've had a couple of careers and ended up as a
| research software engineer working on AI. I often feel nostalgic
| about pure math (maybe even a little regretful I left academic
| math). But I think it'd be almost impossible for me to return to
| academic math. The 3B1B videos always remind me that math is
| available to all and you don't have to be a working mathematician
| to enjoy, learn, and even discover, new math. You don't have to
| be employed as a mathematician in a university.
| noqc wrote:
| The original understanding of a manifold was simply a
| "configuration space", which is very concrete, so I'm not sure
| what you mean that you are surprised that the world of topology
| could be practical.
| jebarker wrote:
| I didn't say I'm surprised the world of topology could be
| practical. I said that _I_ wouldn't have been able to explain
| the concepts in the video so clearly and tie them to a
| practical problem.
| noqc wrote:
| you actually said "I don't think I could have tied the
| esoteric word of topology to a practical problem".
| blueredmodern wrote:
| Is there some area of math that you consider particularly
| useful for software developers?
| jebarker wrote:
| Depending on the area of software development then
| trigonometry, geometry, linear algebra, number theory,
| combinatorics and probability theory are the most obviously
| useful. Beyond that I know that there's a close relationship
| between category theory and functional programming. I'm not
| familiar with the details of that or whether it's useful in
| practice or more of an area of theoretical study. I'm sure
| there's others on HN that know though. Interestingly I used a
| fair amount of category theory in algebraic topology, but
| never closed the loop and learned much about the relationship
| to programming.
| pfdietz wrote:
| > working on AI
|
| I think we're about to enter an incredible new age of
| mathematics, driven by AI and theorem provers. It's going to be
| hugely disruptive to mathematics, but lots of fun to amateur
| mathematicians.
| jebarker wrote:
| Yeah, I really hope so. I'm hoping that my background is
| going to allow me to work/play in this area. I'm currently
| learning about theorem provers so I can get involved.
| vismit2000 wrote:
| Recently on HN:
| https://news.ycombinator.com/item?id=42440016
|
| Maybe this can aid in your learning.
| jebarker wrote:
| This is great - thanks for sharing
| vhxs wrote:
| I agree. My PhD is technically in CS but it made heavy use of
| algebraic topology. Being 5 years out, having worked briefly in
| tech, then at a national lab as a software engineer has given
| me enough outsiders' perspective on pure math. You probably
| need to work as a professional mathematician to be at the
| research frontier of a given area, but otherwise the
| fundamentals of math are unchanging, and in my opinion, that
| makes it accessible to anyone who is sufficiently interested in
| and passionate about math.
| rvense wrote:
| When I was doing my degree (area studies and linguistics), a
| friend who was in mathematics liked to tell me that mathematics
| was the second-most democratic science: all you need is a pen,
| some paper, and a waste paper basket; the humanities were the
| only thing that was more accessible - since we don't even need
| the waster paper basket...
|
| (I also miss my old subjects, not to mention being young and in
| university)
| WhitneyLand wrote:
| Does anyone else feel anxiety watching this? I guess some fear of
| failure/over achiever residual worry hangs on.
| klysm wrote:
| Anxiety about not understanding immediately?
| WhitneyLand wrote:
| Yeah, weird right? It's related to what's sometimes called
| gifted kid burnout.
| klysm wrote:
| Yeah I can relate, high expectations result in
| disappointment eventually
| boothby wrote:
| We generally don't talk about downvotes... but I gotta say that
| it's sad that this comment was gray. Your approach to an
| uncomfortable feeling was to name it, and get curious. That's
| commendable, and you're brave to share such in public. We
| shouldn't be punishing that here. Curiosity, like the response
| of klysm, is warranted.
|
| I've got a PhD in math and I've largely retreated from the
| academic pursuit. The thing that got me through my degree
| wasn't a drive for success or academic attainment, but love of
| the journey. Once I found employment, math turned dark and
| scary to me for quite some time, and this video was a breath of
| fresh air.
|
| I hope you find a source of joy that you can apply yourself to.
| From such a root, you can flourish. It needn't be work, in
| fact, I believe that the perilous job market underlies my
| anxiety. My root is my chosen family, not my career. With that
| security, it's easier to let one's mind wander and pursue
| puzzles like this open problem (should they capture you). But
| it starts with curiosity.
|
| Once, at a conference, John H. Conway admitted to me that he
| felt the very same as you for a period early in his career.
|
| And speaking of failure: I woke up with an idea for how to
| approach the open problem. I hacked up some code to apply my
| approach to the Koch snowflake. In writing it out, I found the
| obvious problem with my approach (context-free punchline:
| spotted the division by zero before I wrote down the line of
| code that would have triggered it). It was fun to fail, because
| nothing depended on me succeeding in that effort. And spotting
| bugs before they're written is always satisfying.
| WhitneyLand wrote:
| Thanks so much for the thoughtful comment. I hope others find
| it as hopeful and motivating as I did.
| madihaa wrote:
| This video has now taught me what topology is.
| klysm wrote:
| 3b1b shows us what's possible in math pedagogy. I'm excited for
| the future of the space, but sad it will take so long to adopt
| methods like this for teaching math
| shagie wrote:
| The amount of effort to do a single 30 minute video of this
| sort when scaled out to a half or full year math class is
| significant.
|
| Another consideration is that we learn things from it because
| we want to learn it. We are engaged with the topic the instant
| we hit play _because_ we want to watch it.
|
| Compare that with a high school or college setting where the
| majority of the class is taking it because they have to - not
| because they want to. This means that there's no initial
| engagement and a professor can't call out the student in the
| 3rd row from the back that is starting to fall asleep.
|
| This can work really well for the people who want to learn it.
| However, it potentially adds to people who don't want to become
| competent in the material falling further behind.
| klysm wrote:
| I agree you can't get around people fundamentally not being
| interested in the material. That being said, I still think
| that the power of 3b1b does should not be understated. It can
| cultivate interest as well!
| FLT8 wrote:
| > The amount of effort to do a single 30 minute video of this
| sort when scaled out to a half or full year math class is
| significant.
|
| This is true if Grant is the only person doing the work,
| however having a well educated and scientifically engaged
| populace seems important enough that we (the human race)
| should devote a few more resources to creating high quality
| (and freely available) courseware for all curricula/year
| levels.
| ninetyninenine wrote:
| I've had classes where I didn't want to learn shit but I
| learned anyway because of videos like this. Like the
| explanation is so clear that as long as you don't fall asleep
| you absorb it.
|
| I didn't become interested in science and math until later in
| my life and I spent much of my childhood in classes where I
| didn't care.
| wruza wrote:
| I usually watch 3b1b without any prior "want to" or any idea
| what it will be about. For me it's the format that drives
| interest.
|
| Although I'm from a natural-math-guy group, in a sense that I
| usually have no issues with understanding the material, in
| contrast to these "so interesting, but I understand nothing"
| comments below it. I always wonder why they watch it, cause
| it must be just a set of vector animations then.
| gf000 wrote:
| High school math has a standardized curriculum that doesn't
| change significantly - it's 100% possible to create this high
| quality material for the whole n years and make use of it
| year after year. Especially that the most important part of
| this is the software used to make these semi-interactive
| graphics (which is open-source), so a teacher can just do it
| their own way, incorporating animations fit for their
| examples - no need to pre-render a video for each day. Do a
| "normal" class and visualize important aspects.
| wbl wrote:
| There is no royal road to geometry just like the route to
| Carnegie Hall is practice practice practice.
| Enginerrrd wrote:
| Yes and no...
|
| Ultimately it would be impossible to come up with these greatly
| simplified explanations without the complexities and notation
| taught in the pedagogy you are dismissing.
|
| Granted though, as students, the gifted ones often already have
| this picture in their mind, so the intuition is obvious. So to
| bring those less gifted or familiar with a topic up to speed,
| things like this make a ton of sense.
| gcanyon wrote:
| I've known about the mobius strip since I was a kid, and the idea
| of existence proofs based on continuous functions having to cross
| since my early teens.
|
| The idea that the mobiu strip is more than a pointless novelty
| has never occurred to me, and now I feel like I have to apologize
| to that object for dismissing it so cavalierly. Its role in this
| proof is remarkable and a wonderful brain tickle.
| edanm wrote:
| If you haven't seen Dr Tadashi Tokieda's lectures on geometry,
| I highly encourage you to watch at least the first one. The
| best introduction to any math topic I've ever seen, I think,
| based around (among other things) the Mobius strip.
|
| https://www.youtube.com/watch?v=SXHHvoaSctc&list=PLTBqohhFNB...
| graycat wrote:
| Another view of _topology_ is in
|
| John L.\ Kelley, {\it General Topology,\/} D.\ Van Nostrand,
| Princeton, 1955.\ \
|
| In the set R of the real numbers and x, y in R with x < y,
|
| (x,y) = { z | x < z < y }
|
| is _open_ and, with x <= y,
|
| [x,y] = { z | x <= z <= y }
|
| is _closed_.
|
| A subset of R that is both closed and _bounded_ is _compact_ , a
| powerful property, e.g., in Riemann integration.
|
| And so forth but in _topological_ spaces much more general than
| the real line and open and closed intervals. Apparently hence the
| "General" in the book.
|
| As a math major senior in college, read Kelley and gave lectures
| to a prof. But now there are some other definitions of
| _topology_.
| devil12gamer1 wrote:
| Love you
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