Can anyone explain to me why topology exists? Is it just an exercise

I mean I googled "what is the point of topology" but it just thought I was asking about adherent points

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With topolocgics, you can calgualate all nugular scients with no difficul. All you have to remember is, that when x is 3, answer is equals not really. For example -3^x~d != pi+5.

Allow me to explain:

Three.

>>8447637
this is what a topology phd sounds like

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>>8447635
>what is the point of topology
An element in a topological space or, more precisely, in the underlying set of that topological space.

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>>8447633
are you implying that mobius strips and klein bottles don't exist?

>>8447652
Can you not read?
>discovering how a non-existant material behaves
>material

The mobius strip might be the behaviour, but if you can parse sentences "non-existant" clearly describes the material.
ie. an infinitely stretchable material which can pass through itself, but cannot be cut or bent sharply.

>>8447652
Only 3D drawings of Klein Bottles exist

>>8447633
low quality b8

>>8447726
Not b8, genuine question.

>>8447728
there are more applications than physical forms, which as you correctly indicated, are limited to 3 dimensions.

you can have a phase-space in a database of interactions, for example, which could lead to an massive reduction in runtimes for algorithms that have to process that data.

>>8447709
woah dude

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>>8447633

object classification,registration,detection,recognition

having a basis to further develop shit in analysis, differential geometry and all that shit

Topology isn't about "stretchy materials".

Topology is just more-general analysis.

The "stretchy materials" are simply the pretty pictures that topologists use to entertain popsci casuals like you

It's taking me an insane amount of time to understand this topology stuff.
Hope that's normal.

>>8447633
No, see fixed-point theorems of any kind, theorems about fiber bundles (and fiber bundles themselves) are important too.

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>>8448000
Draw pictures. It makes several claims pretty intuitive, for example how the Möbius strip can not be embedded into R^2. An example fit for this thread now that it has been mentioned several times.

Fiber bundle with base = disk interior is trivial.

True? Why?

>>8448005
Yes drawing pictures help me.
But that's only when the spaces are "metrique" no ? When it's a topological space and there's no "distance" can you still draw stuff ?

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>>8448028
Draw blobs freestyle. Draw, frenchman, draw! Break free from Bourbaki's anti-picture mentality! I'll give as an example how to imagine that continuity preserves connectedness. You can see X's pain as it gets ripped into pieces by our bad assumption.

>>8448028
CW complexes are another (perhaps even better) approach than just drawings.

>>8447635
You made this whole thread just to post that joke didn't you...

>>8447804
this is the real answer

it makes notions about convergence, filters, connectedness, etc. a lot more general.

so much more general that we can develop spatial reasoning about all sorts of things which aren't traditional spaces.

for example, for any logical theory you can build the stone spaces of n-types. the "level of disconnectedness" of the space tells you a lot about what kinds of elements you might find in different models of the logic theory, and which ones are only in some models. it is sort of a set of "potential elements/tuples" which have been TOPOLOGIZED according to how much the logical theory can tell things apart.

schemes offer a notion of localization more relevant for studying polynomial equations than the traditional topologies.

they also allow you to use basic spatial intuition about stretching, gluing, cutting, etc. to reason about things that would be opaque from an analytic point of view. topology gives you a hint as to whether to even bother with the - usually much more computationally difficult - analytic methods.

and this stuff >>8448002

for example, nash equilibrium can be seen as a direct consequence of some fixpoint theorems, where the existence of fixpoints is implied by a notion of CONTINUITY which is made precise in topology. often the "points" of spaces we are interested in are functions, and convergence in these spaces tells us about limits of functions.

>>8447734
This is terrible, higher dimensions have nothing to do with topology. Higher dimensions are more often studied with geometrical and analytical tools.

To answer OP's question so I don't seem like a complete asshole there are countless applications for example in PDE theory and Modern Physics (knot theory etc.). Too many to name.

There's a homotopy between identity and constant maps on a Euclidean space. I find it nonsensical.

If you want applications of topology to "real world stuff," then physics is a great example. A lot of physical properties of things can be calculated based just on the topology of a system, independent of extra geometry. You have things like Gauss-Bonnet, which essentially say that smooth bending and stretching of a geometric object are independent of the topology. It can greatly reduce calculations. Also in physics, we are often interested in a gauge groups, and the tools of cohomology and homotopy theory simplify and clarify what is going on upstairs. Statements from analysis, such as the intermediate value theorem, are almost trivial to prove using topology.

Also, it turns out that algebra and geometry are "dual" in some meta-formal sense (Jacob Lurie has done a lot of good work to actually formalize this), so loosening geometric structure ends up loosening the correspondence algebraic structure. This can be helpful, as it lets us more clearly see if and why theorems should generalize or not. Moving from discrete spaces to compact Hausdorff spaces moves you from complete atomic Boolean algebras to C* algebras, but we can pretty easily trace what structure remains when we generalize, and then theorems that make no reference to the "extra stuff" in the Boolean algebras can port over to C* algebras. The dictionary formed this way tells us how to then turn the dual theorems into ones for the more general spaces.

Oh thank god an OHP post, I thought I was going to have to type up a serious reply to this shitty thread.

>>8448105
Why? Just imagine the space contracting continuously down to the origin. It's useful, as others have pointed out, in proving fixed-point theorems. A lot of theorems are independent of data that doesn't mess with loops, too. We end up with invariants that are easy to calculate but still carry lots of information.

Topology is necessary for functional analysis and operator theory, which means that it's necessary for qua tum mechanics. There's also emerging applications of topology to data analysis

>>8448116
>Why?
Image. It's the whole space at [0, 1) of homotopy and it's one dot at 1. Hard to believe it's continuous

Topology is necessary for functional analysis and operator theory, which means that it's necessary for qua tum mechanics. There's also emerging applications of topology to data analysis

>>8448134
>Topology is necessary for functional analysis and operator theory,
> necessary

No.

>>8448133
Yes, I suppose it's counterintuitive when first encountered. It's an important step in learning homotopy theory to break your intuition away from continuity to contractions and extensions. If you can nail down the idea that every homotopy equivalence (at least between CW complexes) can be broken down into an embedding followed by a contraction or into an extension followed by a quotient, you should be set.

>>8448149
You realize completeness, compactness, etc. are all topological properties right?

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>>8448133
But the whole space is homeomorphic to the open unit ball, too. Considering this, it is reduced to the ball being contracted to a point.

>>8448010
>True? Why?
This is true. One way to show this is by retracting the disk to a point, which is a continuous map. There should then be a continuous map from the vector bundle to the vector bundle on the point (which is trivial because there is only a single fiber). Such a map gives you a coordinate transformation on every fiber that turns the vector bundle into a direct product.
Intuitively, you can orient the vector spaces at every point in the base space to line up simultaneously.

>>8448162
You realize they have analytical and set theory equivalents rights?

>>8448198
What is that even supposed to mean? Completeness, compactness, etc. are topological properties. Sure they can be given various definitions. i.e. In metric spaces the can defined sequentially, but it is still topology.

>>8447633
Honestly I have no idea why mathematicians study maps and the surface of the earth and shit.

>>8448218
>What is that even supposed to mean?
That their definition as topological properties isn't required since they can be defined without invoking any topological ideas by using ideas from other disciplines.

ie. topology is not necessary for func anal or operator theory.

>>8448225
I don't think you understand how broad a subject topology is.

>>8448227
Motherfucker, you're not getting my point.

Just because it has applications in another field doesn't mean that field requires it.

Let me give a simpler example of the situation
>Calculus can be used to calculate the area under a 1 dimensional smooth curve
>Nuh uh, real analysis is NECESSARY to calculate the area
>But if we only had he calculus idea for a limit we still do it
>DO YOU EVEN FUCKING KNOW HOW BROAD ANALYSIS IS

>>8448231
>Motherfucker
relax edgelord

>>8448232
Fine, you autistic brainlet.

>>8448231
Saying functional analysis doesn't require topology is a stupid as saying it doesn't require linear algebra.

>>8448233
i'm not even the one you were replying to dicklet

you have no understand of mathematical dependence

>>8447661
>an infinitely stretchable material which can pass through itself, but cannot be cut or bent sharply.
What is spacetime?

>>8448255
Spacetime is a smooth manifold and hence cannot self-intersect (unless you want to try and interpret blackholes as self-intersections) . So "pass through itself" wouldn't work.

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>>8447633
>non-existant material behaves

If mathematics exist, the object exists.

>>8448260
What exactly stops a "smooth manifold" form self intersecting? The Klein bottle in OP's picture appears to be smooth and it self intersects. Maybe I'm just confused about what a manifold is.

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>>8448283
It only intersects itself in that pic because the pic has too few dimensions. In 4D it doesn't.

>>8448283
>What exactly stops a "smooth manifold" form self intersecting?

A self-intersection point is a singular point from the view of differentiability.

>The Klein bottle in OP's picture appears to be smooth and it self intersects.

Embedded in 3-space, the KB is not smooth. However it can be smoothly embedded into 4-space in a way that eliminates the self-intersection.

>>8448296
Okay, so you're saying that the infinitely stretchable material does NOT pass through itself. Then topology is totally applicable to spacetime, right?

>>8448299
>A self-intersection point is a singular point from the view of differentiability.
So a smooth manifold cannot have any singular points? Sorry I haven't taken analysis in quite awhile.

>Embedded in 3-space, the KB is not smooth. However it can be smoothly embedded into 4-space in a way that eliminates the self-intersection.
Yeah I know, but the OP claimed that topology is the study of the behavior of material that can pass through itself. Since the KB doesn't pass through itself, are there any topological objects that do?

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>>8448303
Nope, I never said anything like that. I only answered your question regarding self-intersection. It is a question of where to embed the Klein bottle, and with 4 or more dimensions it is like it should be, with less than 4 it can be considered a projection of the 4D version. These projections suck, but the only way for us to visualize it is to take its projection. Similarly, you could make a Möbius strip out of paper and see there are no self-intersections, but, when drawn on paper, there are two distinct points getting "mapped" to the same point (are indistinguishable in the drawing).

How to relate this to spacetime? I have no idea. I'm not a physicist.

>>8448173
Oh, I didn't think the explanation could be so intuitive, now I get it, thank you!

>>8448315
>So a smooth manifold cannot have any singular points?

Correct. If you want to study singular spaces, you can look at orbifolds (or move into algebraic geometry).

>but the OP claimed that topology is the study of the behavior of material that can pass through itself

OP is a faggot

>>8448338
So does the "material" in topology pass through itself ever? Or does this only happen when we take the object into lower dimensions? I realize OP is a faggot but my question is still unanswered.

>>8448315
>So a smooth manifold cannot have any singular points?
Nah, but it's actually in the definition. Every point must have a neighbourhood that is homeomorphic to an open ball, which doesn't allow for it to self-intersect (think about a curve that crosses itself and how it doesn't allow for the intersection point to have a neighbourhood that is homeomorphic to a real interval).
That being said, it's not completely obvious why it wouldn't work, there's some topology involved

>>8448377
Okay, I just didn't know what a manifold was. I probably should have looked up the definition before discussing it. Do you mind answering >>8448353? Is the "material" discussed in topology also considered a smooth manifold?

>>8448399
>Is the "material" discussed in topology also considered a smooth manifold?

Topological spaces are quite general objects. Smooth manifolds are a very special case. i.e. Paracompact Hausdorff spaces locally diffeomorphic to euclidean sapce

>>8448403
Okay cool, thanks.

>>8448423
not that poster, but just to give you an idea of how general topological spaces are, the following are all subcategories of the category of topological spaces with continuous functions:

smooth manifolds (so all euclidean spaces)
the dual of boolean algebras (stone spaces)
all preorders (alexandrov spaces)
all sets (discrete spaces)
spectral spaces (affine spaces associated to commutative rings, e.g. "C with the Zariski topology and a generic point")

so the notion of "space" and "continuity" is one where at least all of the above can be considered spaces, and is accordingly very general.

>>8447635
kek

>>8447633
yes topology has real world applications.

it's enormously important in physics if you believe that is real.

and there is other random shit as well. I remember being at a talk were people are using algebraic topology for camera data compression algorithms.

It's not always apparent when pure autism fields of math have actual applications but they always seem to down the road in strange ways.

Nobody predicted that analytic number theory would have applications to cryptography because computers didn't exist at the time. Hardy literally rolling in his grave.

>>8448435
>Zariski topology
I'm not him, but is it true that topology on algebraic varieties is induced from topology on a spectrum (points of variety viewed as max ideals, open sets are retrieved as intersections of open sets with a those max ideals)? Is it done for any ring?

>>8447709

Weird but true.

>>8448472
yes. the category of affine schemes is trivially dual to category of comm ring with unit. grotheniecks insight is that we take all prime ideals to be points, not just maximal. e.g. spec C[x] will have a "generic point" for 0 ideal. this is necessary, since inverse images of max ideals are not necessarily max, while inv images of prime ideals are always prime. so a ring hom R->S gives a set function from Spec S->Spec R. we can topologize the sets by taking a basis open for each ring element of all points not vanishing for f (so, f not in P, ie f mod P is nonzero). giving the spectra this topology has the result that the induced function is continuous. to get full duality though, we need to equip each spectrum with something called a sheaf, which assigns to each open formal limits of formal fractions of elements of R, where the denominator is nonvanishing on the whole open (so that we don't divide by 0). the sheaf will assign to the whole space the ring R. the continuous functions above actually preserve the germs of formal fractions near each point in the sense that for each preimage of a point, there is an associated homomorphism of local rings.

>>8448497
sorry you asked about varieties. projective varieties may not arise as spectra. however, we can look at a more general class of schemes, which are spaces with sheaves of rings, such that every point has an open neighborhood which is isomorphic to an affine scheme. we can "glue together" affine schemes to construct varieties, though they will have these "extra" nonclassical points for the nonmaximal primes.

>>8447633
Knots and how to tie/untie them.

That's most of the practical use of it.

>>8447633
>material
if your only exposure to a subject is shitty popsci videos, don't talk about it, don't ask about it, just fuck off
wait until your freshman year for real analysis, and then you'll understand what topology is
unless you're an engineer in which case go suck cocks

>>8448519
To get even more weird, there is the notion of analytification. Which is, if you start with a variety, you can construct a corresponding analytic space. For example complex varieties which are given by polynomials over the complex numbers, are very algebraic objects with a course zariski topology. Their analytification puts a complex manifold structure on your variety which gives you all the tools of analysis at your disposal.

These two spaces which live in different worlds are related by an equivalence of categories between their coherent sheaves.

We should start an alg geo thread, this stuff gets my reimann-rochs off.

>>8448688
>if you start with a variety, you can construct a corresponding analytic space.

i.e. Serre's GAGA

Which itself further generalizes to etale topologies on noetherian schemes.

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>>8447637
>when x is 3, answer is equals not really

>>8448283
>4D
It's a 2 dimensional surface...

>>8448065
Mathematics must not be visualized!

>>8448519
> projective varieties may not arise as spectra
If so, what can zariski topology / corresponding scheme say about the algebraic variety? Is there any reason it's considered?

>>8447633
>thinking topology is about how materials behave

You're a fucking idiot. Topology is analysis of spaces, which occur naturally in a variety of ways. The easiest one is the phase space of mechanics which is 6n dimensional where n is the number of bodies considered.

>>8448935
It's a 2D surface in the sense that it is locally homeomorphic to R^2
It's """""4D""""" in the sense that it can be embedded in R^4 and not R^3

>>8448263
>platonic realism
there's the door

>>8447637
this made me giggle

>>8449053
Well, for one, now it's in the category of schemes with all sorts of other algebro-geometric objects, so we can use general techniques on it.

We can find continuously parameterized proj varieties by looking at fibers over another scheme. We can take products with arbitrary other schemes.
Whether it is reduced or separated or... can be described as part of a general phenomena.
We can take fibers over Z to get "mod p" results for arbitrary schemes with a map to Spec Z.

Part of the whole point of scheme theory, like diff geo, is that we can talk about varieties intrinsically, without thinking of what coordinate system they came from etc, and hence describe a lot of properties of them in terms of properties of morphisms.

>>8447734
please refer material. ty

>>8449053
alot of natural varieties we would want to talk about aren't affine. For example diff geo arises because you want to do calculus on things like spheres that are not euclidean.

>>8450069
Affine varieties are definitely not euclidean spaces.

>>8450108
Yes, so what? I guess it was an analogy.
affine <-> euclidean
general varieties <->general manifolds

>>8449065
A 2D manifold embedded in R4 is still 2D...

It is in no way 4D.

>>8449059
>The easiest one is the phase space of mechanics

kek

>non-existent
>Möbius strips and Klein bottles
You're right OP. Flexible materials and glass don't exist.

it is worth noting that the Nobel on Physics was awarded to some nigs for something something topological something

>>8448435

Full on Autism: The Thread
You dont have any idea of how general lingüistics is.

>>8450318
>Nobels

>>8450542
>You dont have any idea of how general lingüistics is

you wat?

>>8450232
That's what the silly excess of quotation marks means newfag.

>>8448248
This

>>8447637
hilarious , can u explain how you make sentences like taht i would like to do it to

Why did the chicken cross the mobius strip?
[spoiler] to get to the same side

Topology is used in combustion, fluid mechanics, optics, qm, etc...

>>8447633
>Can anyone explain to me why topology exists?

The correct question is pronounced:

"What is the practical application of topology"

Pure-Mathfags hate it when you ask them for a practical application of their work.

Might be some applications in architecture for topology, though.

>>8447633
As good a thread as any to ask.

Does anyone know of any software I can use to plot a 2D surface (or an image of one in any format) on a n-connected torus in 3D?

Alternatively how can I convert paths on a 2D plane model (with appropriate identifications) into 3D parametric equations?

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>>8447633
>topology
Here you go:
https://en.wikipedia.org/wiki/Topology

>>8453242
What the fuck is your problem Anon?