What is a Topological Space?
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- čas přidán 11. 04. 2020
- Introductory video on topology that explains the central role of topological spaces in mathematics. Examples include indiscrete and discrete topologies on a finite set X = {1,2,3}
Recommended resource for topology:
* Elementary Topology Problem Textbook (O. Ya. Viro, O. A. Ivanov, N. Yu. Netsvetaev, V. M. Kharlamov)
Link: www.pdmi.ras.ru/~olegviro/topo... - Věda a technologie
The fundamental law of math: If you think you understand a topic, there's always a more difficult generalisation of that topic you don't understand
(At least for me)
But thanks for the link to the textbook, now I know what I'm doing this summer holiday (in Aus)
lmao underrated
This OOP example really hit, thanks a lot
What an articulate and beautiful presentation
Thanks. The video is quick but informative, it gives a nice overview.
this was very clear and helpful, thank you!
Well done! I understood everything on the first try, love it!
Many thanks for the quick informative overview. I have following remark: because a scalar product induces a norm, the shown diagramm can be extended with one more cycle.
very great explanation! Thank you for that content
good video; however, not all vector spaces are topological spaces, but they can all be given a topology. Same for metric spaces, but they have a nice induced topology.
Hey I have a question
Great job - very clear
what an amazing video. keep doing it. thank u always 💛
you clarified a lot for me. thanks a lott
Great video! Could you do one about generators of sigma algebras too?
Thank you so much. It made it click right away. :D
Excellent video
Very clear Thank You
Please correct me if I'm wrong, but I don't think all vector spaces are metric spaces. Only normed vector spaces.
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.
Very Clear!
- Excellent.
- Thx.
Nice explanation
MAD RESPECT.
Thank you
It was useful
Where are you? Keep it up, you are the best
Thanks man, much appreciated. I'm currently at UWaterloo doing my masters in Applied Math. I'm planning to drop more topology videos soon!
Thanks fam.
Nice overview
The picture, which is included as ta part of this CZcams 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 CZcams creator had something else in mind, but....the fact, as indicated, remains.
you're diagram is quite confusing as if you are saying that every vector space is a metric space which is false.
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.
@4:45 why is intersections considered for a finite number of sets but union for an infinite number of sets?
A great answer can be found here: math.stackexchange.com/questions/284970/in-a-topological-space-why-the-intersection-only-has-to-be-finite
Great video. Only suggestion is to buy a mic.
Good
I am not sure about this video. You started comparing to subsets of animals that they inherit everything from the bigger set. Then you said vectors, which are a subset of metric space have more properties then Metric space.
How could a union of subsets in tao not itself be in tao?
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.
I will to this day never understand the need, other than as a mere interesting curiosity, of putting a topology onto uncountably infinite groups, such as Lie Groups.
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.
*object* : that which you can imagine or draw; that which has shape
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.
My attitude may be too nitpicky for the general audience you are targeting with this video, but I don't think a vector space, as usually defined from the axioms, is a good example of a space to start with. It is more like an algebraic structure, it doesn't in itself have space-like features like distance or nearness. Now a k-vector space over a specific field k (like the real or complex numbers) may inherit such space-like features from its underlying field (a space-like real line or Argand plane). But the mathematical properties of a vector space itself (closed under linear combinations, spanning sets, choice of basis, matrix representations) aren't really space-like. The so-called vector "space" is more of an algebra-like coordinate system that you layer on top of some underlying genuinely space-like thing (such as the space that is the product of two or three non-parallel real lines). Admittedly, most calculus students will identify the algebra-like coordinate system with its underlying metric space with the Euclidean notion of distance, but I think that can be a source of confusion since such students would also be exposed to matrix-based linear algebra and programming over coordinate systems. I'm not a mathematician, my background is in software engineering, but I think it is better to think of the vector space as a coordinate system, a discrete thing with algebra-like operations, distinct from an underlying space or product space which is what has space-like features such as distance and continuous functions. Importantly, a vector space isn't a special kind of topological space (so the analogy with object-oriented programming and animals breaks down), only certain vector spaces over a space-like ground field "inherit" the features and mathematical properties of a topological space. A vector space over a finite field (such as a number system of modular integer arithmetic, like clock arithmetic modulo 12 or 24) is a perfectly legitimate vector "space" that isn't space-like, it is very discrete. Think of the discrete vector space generated by two digital clocks, one giving 12 hour numbers, the other giving military-style 24 hour numbers. This is pretty intuitive, you can program it just like other vector spaces, and do all the standard linear algebra like things including Gaussian elimination of matrix rows, but it is not a special kind of topological space or metric space. You can't do calculus over it or have topology-like continuous deformations. (Technically, I guess you can define a discrete topology over the (power set of the) set of number pairs, but that is structure beyond the vector space itself, and doesn't really support intuitions about spaces, it is one of those notorious mind-twisting abstract topology examples, a degenerate discrete topological structure).
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
topology and abstract math is like opening the elder scroll from skyrim wtf is this shit lmao
Mnmnn😊
A general vector space is NOT a metric space!! There is no canonical metric on it!
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 👍🏼
This video confused me more. I'm trying to be very precise when I learn things, and as mentioned, a topological space is not entirely a generalization of metric/vector spaces.
Also, it would be more clear with the definition to say "if an arbitrary set of elements are in tau, then their union is also in tau." You simply said the latter piece, "arbitrary unions are in tau," without first stating that the sets that form the union are also in tau.
You also refer to the pieces of the definition (5:29) as "the first axiom." These are not axioms, they are pieces of the definition.
Being precise with language is important in math even if "just to give intuition."
Also, need to give more pauses between sentences so we can absorb what you said.