Can someone please explain to me what topology is?

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    Explain Topology
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Discussion Overview

The discussion revolves around the concept of topology, exploring its definition, intuitive understanding, and its relation to continuity and geometry. Participants share various perspectives on what topology entails, including its applications and educational context.

Discussion Character

  • Exploratory
  • Technical explanation
  • Conceptual clarification
  • Debate/contested
  • Mathematical reasoning

Main Points Raised

  • Some participants describe topology as the study of the geometry of continuity, questioning the meaning of continuity itself.
  • Others mention that topology is often referred to as "rubber-sheet geometry," emphasizing the idea of stretching and compressing shapes while maintaining their topological properties.
  • A participant suggests that a human can be considered topologically equivalent to a torus when plugging their nose, leading to humorous exchanges about the implications of this analogy.
  • There are discussions about when topology is typically taught in academic settings, with varying experiences shared regarding the timing of courses in different educational systems.
  • One participant elaborates on the definition of continuity and its relation to topology, mentioning concepts like neighborhoods and homeomorphism.
  • Another participant challenges the idea of being a double torus, questioning the number of holes in the human body and the implications of plugging one's nose.
  • Some participants engage in light-hearted banter about the anatomical analogies, including references to orifices and digestive tracts.

Areas of Agreement / Disagreement

Participants express a range of views on the definition and implications of topology, with no clear consensus on the analogies used or the specifics of topological equivalence. The discussion remains unresolved regarding the anatomical interpretations and their topological significance.

Contextual Notes

The conversation includes various assumptions about continuity, dimensionality, and the nature of topological properties, which may not be universally agreed upon. The playful nature of some analogies may obscure more technical discussions.

Who May Find This Useful

This discussion may be of interest to students or individuals curious about the foundational concepts of topology, its applications in mathematics, and its educational context.

tongos
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Can someone please explain to me what topology is?
thanks
 
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For these types of inquiries, I refer to google and wikipedia.

An intuitive explanation can be found http://www.shef.ac.uk/nps/Wurble.html .
 
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it is the study of the geometry of continuity. do you know what continuity means?

if so can you see that, from the point of view of continuity, a circle and a line are different, but a circle and a square are the same? and a line and a half circle are the same?
 
Similarly, a human is topologically equivalent to a torus when he plugs his nose. :smile:
 
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i appreciate it.
My grandpa said it was like geometry shapes curved on spheres, like rubber geometry. And you deal with dimensions. Is this true?
 
ya topology is also nicknamed "rubber-sheet geometry" because if you imagine everything as being made out of rubber you could stretch/compress it into other shapes "homeomorphic" (similar-shape) to the one you start with. a topologist is someone who can't tell the difference between a doughnut & a coffee cup
 
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the coffee cup must have a handle , if it doesn't any topologist can differentiate it from a doughnut :rolleyes:

-- AI
 
what if it were an english doughnut? (no hole, jam in the middle, only calling it english to differentiate it from more familiar to some items).
 
matt grime said:
what if it were an english doughnut? (no hole, jam in the middle, only calling it english to differentiate it from more familiar to some items).
we have something like this, and it's called "sufgania" (this is how you pronounce it in hebrew), but it's too much fattening, beware!
 
  • #10
When does one usually take topology? In college?
 
  • #11
I took it (point-set topology) second term of my second year (right after real analysis). I don't ever recall seeing algebraic topology before final year, though I could be mistaken.
 
  • #12
Depends where you are. Proper topology is taught in the 2nd/3rd year of a good university (undergrad) in the UK. or the 4th year of some, and is possibly taught at US at 4th year level at a good university (eg chicago) though it is more often left to a grad school level.
 
  • #13
Sirus said:
Similarly, a human is topologically equivalent to a torus when he plugs his nose. :smile:
No, then we are topologically equivalent to a DOUBLE torus.

(You've forgotten the big chute connecting your (open) mouth and ass)
 
  • #14
arildno said:
No, then we are topologically equivalent to a DOUBLE torus.

(You've forgotten the big chute connecting your (open) mouth and ass)

That reminds me of a picture and a small discussion in "A Brief History of Time" by Hawking.He gave a reasonable explanation why a dog could not exist in 2 dimensions :-p .I believe it could be said for every living organism with a digestive tractum.

Daniel.
 
  • #15
dextercioby said:
That reminds me of a picture and a small discussion in "A Brief History of Time" by Hawking.He gave a reasonable explanation why a dog could not exist in 2 dimensions :-p .I believe it could be said for every living organism with a digestive tractum.

Daniel.

Why couldn't a dog exist in 2d ?

it just so happens we live in 3d ? nobody knows why ?

roger
 
  • #16
For it to exist in 2D,it must have another internal structure,i.e. the absence of the digestive tractum.Only holes that do not communicate.If it had a digestive tractum,then the dog would be made up of 2 independent areas (like 2 disjoint sets/domains) and therefore it would have to be made up of 2 independent halves of a dog.It would be just like in that sadistic joke.

Daniel.
 
  • #17
the first encounter with topology occurs when one first has a proper definition of continuity, which may occur in the first year of calculus. the first theorem may be something like: "an increasing sequence of real numbers has a (finite) real limit if and only if the sequence is bounded above".

topology has already begun to appear in the definition of continuity, if one has the concept of a "neighborhood" in the definition. i.e. a neighborhood of a point p, is a set of reals which contains an interval of form (p-e,p+e), for some positive e. then a function defined on the reals is continuous at p, if for every neighborhood V of f(p), there is a neighborhood U of p, such that f(U) is contained in V. This definition of continuity uses only topological properties.

once we have continuity we can define homeomorphism. i.e. two sets are homeomorphic if there are mutually inverse mappings between them, both continuous. then topology is the study of properties which are the same for any two homeomorphic sets. for example in the reals, a set S is homeomorphic to an open interval T if and only if S is also an open interval.
 
  • #18
arildno said:
No, then we are topologically equivalent to a DOUBLE torus.

(You've forgotten the big chute connecting your (open) mouth and ass)

I don't understand. A torus has one hole in it. If you plug your nose, the only hole in your body is this very 'chute' you speak of. To be a double torus, there would have to be two holes. What is the other hole?
 
  • #19
Sirus said:
I don't understand. A torus has one hole in it. If you plug your nose, the only hole in your body is this very 'chute' you speak of. To be a double torus, there would have to be two holes. What is the other hole?

your arm is a handle isn't it? it would be attached to your shoulder at 1 end, & your nose at the other end, and that makes a handle. so there's one hole between your arm & body, and that other hole
 
  • #20
it seems rather obvious there are at least 9 orifices in the average male body.
 
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  • #21
fourier jr said:
your arm is a handle isn't it? it would be attached to your shoulder at 1 end, & your nose at the other end, and that makes a handle. so there's one hole between your arm & body, and that other hole

Wow...missed that. I think the topological human becomes more complicated considering that there are two 'exits' in the digestive tract. Not sure what that makes then.

Anyways, I think we have sufficiently answered the original poster's question.
 
  • #22
but are both exits connected directly to entrance?
 

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