Explaining topology to non-mathematicians

  • Thread starter octol
  • Start date
  • Tags
    Topology
In summary, topology is the study of limits and local properties. It is the most general object in which you can do calculus and includes all limits, including continuity. Topology is important because it helps connect many branches of mathematics and is the basis for understanding smooth changes and continuous functions. It has been used to restate and reinvigorate parts of mathematics and has applications in physics, such as in understanding black holes. Topology is often misunderstood as being related to topography, but it is actually a fundamental concept in mathematics.
  • #36
back to the beginning, topology is useful for proving that there are infinitely many primes!

consider the following topology on the integers. for [tex]a, b \in \mathbb{Z}, b>0[/tex] set [tex]N_{a,b}[/tex] = {[tex]a + nb | n \in \mathbb{Z}[/tex]}.

Each set [tex]N_{a,b}[/tex] is a 2-way arithmetic sequence. a set G in this topology is open if either G is empty or if for every [tex]a \in G[/tex] there exists some b>0 with [tex]N_{a,b}[/tex] a subset of G. (not hard to check that unions & finite intersections are still open, and that Z & the empty set are all open so this makes a topology on the integers)

2 facts:
1) any non-empty open set is infinite
2) any [tex]N_{a,b}[/tex] is closed also. since [tex]N_{a,b} = \mathbb{Z}[/tex] \ [tex]\cup_{i=1}^{b-1} N_{a+i,b}[/tex] [tex]N_{a,b}[/tex] is the complement of an open set it's closed

now to use primeness. since any number except -1 or 1 has a prime divisor p & is therefore in [tex]N_{0,p}[/tex] we get that [tex]\mathbb{Z}[/tex] \ {-1,1} = [tex]\cup_{p\in\mathbb{P}}N_{0,p}[/tex]

if the set of primes were finite, then [tex]\cup_{p\in\mathbb{P}}N_{0,p}[/tex] would be a finite union of closed sets & therefore closed. thus {-1,1} would be open, contradicting 1) above. thus there are infinitely many primes.
 
Last edited:
Mathematics news on Phys.org
  • #37
Very interesting but not so important for Physicists.
 
  • #38
I would think Euclid's proof would be sufficient.
 
  • #39
Maxos said:
Very interesting but not so important for Physicists.

is it not? surely it is good to know things like this since ther are links between random matrices (spectral stuff) and the riemann zeta function which is very heavily dependent on prime numbers.
 
  • #40
Yes you are right.
 
  • #41
Ahem! Back to the question of how to explain to NON-mathematicians. My experience is that it is useless to describe concepts with which we are already familiar; their eyes just glaze over. Simpler is to motivate it with the 7 bridges of Konigsberg. Then state that topology allows one to prove that the goal of crossing each bridge exactly once is impossible, so no one wastes time trying. Everyone understands about the value of not wasting time.
 
  • #42
toplogy? i thought that was elementary graph theory: it contains 3 odd nodes or is it only 1?
 
  • #43
Again, right.
 
  • #44
matt grime said:
toplogy? i thought that was elementary graph theory: it contains 3 odd nodes or is it only 1?

it's a topological thing because the solution doesn't depend on the distances of the bridges from each other, nor the lengths of the bridges. the only thing to be concerned with is the connectivity properties. that's what the wikipedia thing said anyway.
 
  • #45
Hmm, most odd. It is only a personal view but I've never considered graphs as anything other than combinatorics. Indeed I have never seen a graph theory theorem refer to any topology (perhaps the author is using it a nontechnical sense?) of the graph (ie not defining open subsets of the graph). Labels are largely unhelpful I admit, and one can certainly use topological ideas in graph theory as one can in many parts of mathematics.

Part of graph theory is concerned with "rigidity" and if a graph may be embedded in the plane.

I really can't agree with it, on reflection. I can't disagree with the assertion that graph theory is not bothered with how things are embedded in some space, merely the data of vertices and edges, but I don't see how that makes it topology. It seems to merely that it is merely a statement that we have realized these problems can be described in mathematical terms.

If you look at the graph theory (mathematics) link at the bottom of the page you'll find that the word topology doesn't appear at all in the description.
 
<h2>1. What is topology?</h2><p>Topology is a branch of mathematics that studies the properties of geometric objects that do not change when they are stretched, twisted, or bent, but not torn. It is often described as the "mathematics of continuity" and is used to analyze the shape and structure of objects.</p><h2>2. How is topology different from geometry?</h2><p>While geometry focuses on the exact measurements and angles of objects, topology is more concerned with the overall shape and structure of objects. It looks at the properties that remain unchanged even when the shape is distorted. In other words, topology is more interested in the "big picture" rather than the precise details.</p><h2>3. What are some real-world applications of topology?</h2><p>Topology has many practical applications, such as in physics, biology, computer science, and engineering. It is used to study the properties of networks, the behavior of fluids, the structure of proteins, and the shape of space-time in physics. It is also used in data analysis and image recognition in computer science.</p><h2>4. Is topology a difficult concept to understand?</h2><p>Topology can be a challenging subject for non-mathematicians to grasp, as it deals with abstract concepts and requires a good understanding of mathematical concepts such as continuity, sets, and functions. However, with the right explanation and examples, it can be made more accessible to those without a strong mathematical background.</p><h2>5. How can topology be explained to someone without a math background?</h2><p>Topology can be explained by using real-world examples and visual aids. For instance, a rubber band can be used to demonstrate the concept of stretching without tearing, and a coffee cup and donut can be used to illustrate the idea of topological equivalence. It is also helpful to break down the complex concepts into simpler ideas and to use everyday language instead of mathematical jargon.</p>

1. What is topology?

Topology is a branch of mathematics that studies the properties of geometric objects that do not change when they are stretched, twisted, or bent, but not torn. It is often described as the "mathematics of continuity" and is used to analyze the shape and structure of objects.

2. How is topology different from geometry?

While geometry focuses on the exact measurements and angles of objects, topology is more concerned with the overall shape and structure of objects. It looks at the properties that remain unchanged even when the shape is distorted. In other words, topology is more interested in the "big picture" rather than the precise details.

3. What are some real-world applications of topology?

Topology has many practical applications, such as in physics, biology, computer science, and engineering. It is used to study the properties of networks, the behavior of fluids, the structure of proteins, and the shape of space-time in physics. It is also used in data analysis and image recognition in computer science.

4. Is topology a difficult concept to understand?

Topology can be a challenging subject for non-mathematicians to grasp, as it deals with abstract concepts and requires a good understanding of mathematical concepts such as continuity, sets, and functions. However, with the right explanation and examples, it can be made more accessible to those without a strong mathematical background.

5. How can topology be explained to someone without a math background?

Topology can be explained by using real-world examples and visual aids. For instance, a rubber band can be used to demonstrate the concept of stretching without tearing, and a coffee cup and donut can be used to illustrate the idea of topological equivalence. It is also helpful to break down the complex concepts into simpler ideas and to use everyday language instead of mathematical jargon.

Similar threads

  • General Math
Replies
4
Views
914
  • General Math
Replies
3
Views
988
  • General Math
Replies
25
Views
3K
Replies
3
Views
1K
Replies
4
Views
926
  • Topology and Analysis
Replies
8
Views
1K
Replies
33
Views
5K
Replies
11
Views
20K
Replies
157
Views
15K
  • STEM Academic Advising
Replies
3
Views
1K
Back
Top