Відео

Its the most abstract, the most general

Mathematics is fundamentally an escalation of "rules" (or definitions): the fewer rules you have, the more things you can describe. Say for example I state "All things made of organic matter": this would include you, me, your dog and plants. This also encapsulates a smaller group with the additional rule: "All humans". A topological space is a set of very minimal rules, so many other spaces can be described by it.

Putting topologies on groups like the circle group allows you to study the dynamics of different maps on them. For example, whether the times two map on the circle is mixing or not. Is the dynamics of the shift map on the space of sequences over a finite alphabet an ergodic system? None of these ideas make sense without defining basic topologies.

Just to give it a prospective, think about what is a function. But not all relations can be described by functions. For example a one to many relation is not a function, but it is a valid map. Topology has the tool set and methodology to deal with maps. In that sense is more like a Swiss Army knife. Calculus and Vector spaces are great carving knifes, but cannot clip your nails or pick a tooth...so to speak.

We can argue definitions all we want, but check out this article: www.ncbi.nlm.nih.gov/pmc/articles/PMC6539555/ Here's a direct quote from the article (which has 17 citations as of today): "Any vector space is a metric space, as it is possible to compute the distances between instances using any common distance metric such as the Euclidean distance. There are some data sets for which no vector representation is known (e.g., proteins); however, it is possible to compute their distance. Thus, all vector spaces are metric spaces, but the reverse is not true."

@@infinitedimensions9436 A (finite dimensional) vector space is METRIZABLE. This is a big difference to being a metric space. It means you CAN put a metric on it, but you can do so in many non-equivalent ways. Consider the vector space of polynomials. If this is a metric space, then please tell me the distance between the polynomial 3x+1 and the polynomial 7x^2-x? Or consider the vector space of functions. What is the distance from sin(8x) to exp(x)? The article you cite looks .... random. Why not just read wikipedia? First of all, its not a math paper and the quote appears to be nonsense. Second, the statements we are talking about you look up in textbooks, not in research papers, because their definitions have been settled almost a century ago.

@@infinitedimensions9436 Having said that, I otherwise enjoyed the video and just wanted to point out this minor mistake. Keep it up 👍🏼

A great answer can be found here: math.stackexchange.com/questions/284970/in-a-topological-space-why-the-intersection-only-has-to-be-finite

Here's an example: Let's take the 3-element set {a, b, c}, and take the following set of subsets: tau = { emptyset, {a}, {b}, {a,b,c} } This satisfies the first and third axiom (since all the empty set and the whole set are in tau, and the interesections are in tau). However take the union of {a} and {b}, and we get the set {a, b} which is not in the original set of subsets. Therefore tau cannot be a valid topology in this case.

@@infinitedimensions9436 Thank you.

Indeed

A slightly more categorical implementation of this approach is to use Grothendieck universes, a bit like adding a few large cardinal axioms. There is a somewhat advanced category theory video by Mike Shulman that introduces Grothendieck universes in its early part ua-cam.com/video/9MLQDOpy190/v-deo.html. Or you can do what the Stacks projects does, and just assume that a few specific large categories exist, if that is all you need in the math you are developing. No need to solve the problem in general, it might not be the right place and time to address such general issues.

@@fbkintanar Thanks!

I think, adding the structre of a scalare product to a vector space one can make a metric space out of it. Because a scalar product induces a norm. Norm induces a metric.

@paulu_ You are correct. Vector spaces are algebraic structures. Topology, metric, norms, inner products are analytic structures. The former lets you do algebra, the latter lets you do analysis. Normed vector spaces are when you have both, and they are compatible in some sense.

No. I think there's a substantive point to be made here regarding (for lack of a better word) containerization in mathematics. Namely, you start with sets but if you want a mathematical object that includes "all sets", you need a new mathematical construct or else you run into a contradiction. Similarly with small/large categories (i.e. you can create a category that includes all small categories, but it itself is no longer a small category). I admit perhaps we're sweeping the difficulties under the rug, because what kind of mathematical object can include all large categories?

@@infinitedimensions9436 Yeah I was wondering the exact same thing. We're playing a semantic trick IMO in the sense that we're just moving up the problem one level.

@@alganpokemon905 Not really. A set and a category would have two different natures. Something cannot contain itself, thus a "set of all sets", which would be a set, meaning that it contains itself, cannot exist.

Not at all. Of course, you can always define your own formalism if you like, but here you're off to an exceptionally bad start. Why? Because you will then be obliged to define what you mean by "imagine," "draw," and "shape." None of these lead to simpler primitives, as we would expect from a formal system, but instead they call on even more complex and entailed terms that will have to be defined in turn. For "imagine" you will have to give a formal treatment of what it means to be a cognitive agent. For "draw" you will have to give a suitable abstraction for all the classes of media on which drawing may take place. Are you only drawing on flat surfaces, or would curved surfaces, manifolds, hypersurfaces also qualify? Does the act of drawing transform the surface or leave it invariant? And what about "shape"? Ordinarily we think of shape as geometrical, but this would mean that your definition of "object" would be constrained to geometrical forms only, which is not the general notion of object that we would find useful. For example, if we're talking about nouns as objects of discourse, does it really make sense to impose a geometrical shapes on them? Go back and try again. Try to state your definition in simpler, more reducible terms, so that the definition will eventually bottom out.

@@starfishsystems Your argument can be reduced to "nuh uh". Would you accept "that which as more useful than @Dan Razzell" as a definition of object?

@@oflameo8927 "Your argument can be reduced to 'nuh uh'." No, their argument does not simplify to "nuh uh". Either you do not know their point, or you are making a straw man here. "Would you accept 'that which as more useful than @Dan Razzell' as a definition of object?" That is irrelevant, and I sensed some smokes from a red herring. Their entire point was the definition that you gave sucks because your definition is incomplete. What is a woman? Moreover, "that which ..." is obviously not a definition of an object because it suffers the same problem of the definition that you gave.

The difference is, I am actually trying to teach mathematics in a way that is clear and instructive. Here you've name-dropped a bunch of convoluted examples but it does not add any clarity or help students. You claim that "most calculus students" will be confused by geometric intuition, and your suggested alternative is to bring in modular groups for some reason. Given that students have many years of geometric intuition starting from high school courses in physics/geometry, and groups are presented only in second-year university (for most math majors), standard vectors are a more accessible example for students. Frankly, your response muddies the waters on numerous points. You alternate between technical language and hand-waving. You object to my claim that vector spaces are special cases of topological spaces, but only justify your claim using vague language such as "genuinely space-like thing" and "very discrete".

@@infinitedimensions9436 I applaud any efforts to build on geometric intuitions when teaching calculus and analysis more generally. A good example of this is 3Blue1Brown. However, when introducing ideas of topology, I think it is worth being careful about what is the underlying geometry, and what is the choice of a coordinate system. The old physics of Newton and Euler got away with conflating the two, and that supplies a flood of examples for freshman calculus. But a modern outlook on physics (and I might add quantum computing) makes the distinction between the theory as a representation and the underlying physical phenomenon which is independent of the choice of coordinate system. I am simply suggesting that a good time to introduce that distinction is when you are clarifying the somewhat advanced notion of a topological space. You don't have to bring in modular groups, I didn't. I only mentioned modular integer arithmetic, which students learn from their high school programming subjects which never mentions groups. And I am not suggesting you incorporate the idea of groups in your videos, I only mentioned it as a counter example to your slightly "hand-wavy" statement that vector space concept "inherits" from a topological space when it doesn't. (I know, it inherits from a normed vector space...) I sincerely hope you continue to make videos, this is on the whole a good one. And if its presentation invites discussion about finer, nitpicky points, that's a good thing, right? I happen to believe the issue at hand is genuinely important for building the right geometric intutions, its not *just* nitpicky. It is something I struggled with and am still struggling with in trying to understand the uses of topology (and algebraic topology even) in computer science and logic. There is no need to get defensive about your choices in making this video, they are reasonable choices. It's just that other choices are possible, and I think they are worth commenting on.

I think you should stick to swe 😂

​@@infinitedimensions9436I think fbkintanar is maybe trying to say that the thumbnail is a bit misleading since a vector space is not necessarily a metric or even a topological space. So maybe replacing that example with e.g. Inner product space would have been good. Otherwise good video!

I got a question

Thanks man, much appreciated. I'm currently at UWaterloo doing my masters in Applied Math. I'm planning to drop more topology videos soon!

I'm using Microsoft whiteboard with a Wacom tablet.

Hey I have a question

Alright, so replace “vector space” with “Normed vector space” and the inclusion holds.

@@infinitedimensions9436 That does work, but the space-like properties come from the norm, not the vector space structure.

​@@fbkintanar because a scalar product induces a norm, the shown diagramm can be extended with one more cycle.

The picture, which is included as ta part of this UA-cam video, viz, the one which seems to suggest that 'Vector Spaces' are a subclass of 'Topological Spaces', is misleading, as it, just, is NOT the case. Perhaps the UA-cam creator had something else in mind, but....the fact, as indicated, remains.

lmao underrated

The more you know, the more you know that you don't know.