Set theory in Munkres Topology

[Diffy]

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.

 

[matt grime]

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.

 

[Diffy]

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.

 

[Hurkyl]

That's what I'm doing wrong; I don't drink coffee!