Set theory in Munkres Topology

In summary, Munkres defines the Cartesian product AxB to be all (a,b) such that a is in A and b is in B, and considers it a primitive way of looking at things. He then defines it to be {{a},{a,b}}, and explains that if a = b, then {a,b} will just be {a,a} = {a}, resulting in {{a}}. The need for the {a,b} notation is to distinguish between (a,b) and (b,a) due to the notion of ordering, as sets do not have any inherent ordering of elements. While the proposed definition of {{a},{b}} would result in the same sets for (a,b) and (b,a
  • #1
Diffy
441
0
In Munkres' Topology he defines a Cartesian product AxB to be all (a,b) such that a is in A and b is in B. He says that this is a primative way of looking at things. And then defines it to be {{a},{a,b}}

He says that if a = b then {a,b} will just be {a,a} = {a} and therefore will only be {{a}}.

What I don't understand is the the need for {a,b}, why not just define the Cartesian product to be {{a},{b}}. If a = b you get the same result.
 
Mathematics news on Phys.org
  • #2
You want to distinguish between (a,b) and (b,a). Your idea does not do this.

The key thing is the notion of ordering. Sets do not come with any order on the elements. This is why you need this fiddle if you wish to define the cartesian product of sets purely in set theoretic terms.
 
  • #3
Thanks so much!

So (a,b) = {{a},{a,b}} and (b, a) = {{b}, {b,a}}

Clearly (a,b) != (b,a)

Where as if you use my proposed definition you would have...

(a,b) = {{a},{b}} and (b,a) = {{b}, {a}}, but these are the same sets.

What is interesting is to me now is looking at the intersection and union of (a,b) and (b,a)

The intersection is {{a,b}}, the union is {{a},{b},{a,b}}. Cool!

On a side note Matt Grime, I don't know the correct quote in you signature, but you are clearly missing another essential tool of a mathematician... Coffee.
 
  • #4
That's what I'm doing wrong; I don't drink coffee! :tongue:
 

1. What is set theory in Munkres Topology?

Set theory in Munkres Topology is a branch of mathematics that deals with the study of sets and their properties, specifically within the context of topology. It is used to define and analyze topological spaces, which are mathematical structures that describe the properties of objects in terms of their spatial relationships.

2. What is the difference between an open set and a closed set?

In Munkres Topology, an open set is a set that includes all of its limit points, while a closed set is a set that contains all of its boundary points. This means that an open set does not contain any of its boundary points, while a closed set does. In other words, an open set has no edges, while a closed set has edges.

3. How does set theory relate to topology?

Set theory is essential to understanding topology because it provides the mathematical foundation for defining and analyzing topological spaces. It allows us to define sets, functions, and other mathematical concepts that are used to describe the properties of topological spaces.

4. What is the axiom of choice in Munkres Topology?

The axiom of choice is a fundamental principle in set theory that states that given a collection of non-empty sets, it is possible to choose one element from each set to form a new set. In Munkres Topology, the axiom of choice is often used to construct topological spaces by choosing a subset of a given set to be the set of points in the space.

5. What are the benefits of using set theory in Munkres Topology?

Set theory provides a rigorous and formal language for defining and analyzing topological spaces. It allows us to prove theorems and make precise statements about the properties of these spaces. Additionally, set theory provides a common framework that allows for the comparison and connection of different topological spaces, leading to a deeper understanding of the subject.

Similar threads

  • Math Proof Training and Practice
Replies
1
Views
920
  • General Math
Replies
8
Views
2K
Replies
4
Views
492
  • General Math
Replies
1
Views
892
Replies
3
Views
1K
  • Topology and Analysis
Replies
2
Views
1K
  • Set Theory, Logic, Probability, Statistics
Replies
1
Views
1K
Replies
2
Views
236
  • Set Theory, Logic, Probability, Statistics
Replies
3
Views
1K
Back
Top