Understanding the product topology

In summary, the problem entailed trying to visualise the product topology between two metric spaces and not being sure how to conclude that the topologies are the same.
  • #1
tomkoolen
40
1
I am having some trouble visualising the following problem and I hope someone will be able to help me:

Let (X, dx) and (Y,dy) be metric spaces and consider their product topology X x Y (T1) and the topology T2 induced by the metric d((x1,y1),(x2,y2)) = max(dx(x1,x2),dy(y1,y2)) so the maximum of their respective metrics.

I have to show that the topologies are the same. I understand that I have to show that the basis opens of T1 should be unions of basis opens of T2 and vice versa.

T1 => T2 is not giving me a hard time: If (x,y) is in U x V, with U open in X and V open in Y, there exist open balls respectively for x in U and for y in V. The minimum of these radii give a ball for (x,y) in U x V. I know that this is the way and I know how to write it down mathematically correct but I am not sure how to conclude that U x V is in T2 now.

T2 => T1 is giving me more struggles. I can create an open ball around (x0,y0) but then I can't visualise the path my solution should follow.

If someone could give me some advice, I would be very grateful!
 
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  • #2
What do open balls in ##(X\times Y,d)## look like? For example, what do the open balls look like in ##\mathbb{R}^2##?
 
  • #3
Thanks for replying! In R2 I could just visualise them as circles around a point (x,y) right?
 
  • #4
No, not with this metric.
 
  • #5
I really don't see it. Of course the metric maps R x R ---> R, for example the distance between (1,2) and (2,4) would be 2 but I don't know how to "draw" it. This is the part where I need a hint.
 
  • #6
Oh I think I should think about the set of all elements of R2 that have fixed distance from (x0,y0) and draw that right?
 
  • #7
What are the points a distance of 1 away from (0,0)?
 
  • #8
{(a,1), (1,b), (c,-1), (-1,d)} with a,b,c,d smaller or equal to 1?
ABsolute value!
 
  • #9
So what figure does that form if you draw it?
 
  • #10
Two line segments of length 2, a "cross"
 

FAQ: Understanding the product topology

What is the product topology?

The product topology is a mathematical concept that is used to describe the topology of a product space, which is the topological space formed by the Cartesian product of two or more topological spaces.

How is the product topology defined?

The product topology is defined as the finest topology on the product space that makes all of the projection maps continuous. In other words, it is the smallest topology that contains all the open sets of the individual spaces and makes the projections from the product space to each individual space continuous.

What are the key properties of the product topology?

The product topology has several key properties, including that it is a Hausdorff space (meaning that any two distinct points have disjoint neighborhoods), that it is a Tychonoff space (meaning that it satisfies the Tychonoff theorem, which states that any product of compact spaces is compact), and that it is a normal space (meaning that any two disjoint closed sets can be separated by disjoint open sets).

How is the product topology different from the box topology?

The product topology and the box topology are both ways of defining a topology on a product space, but they differ in how they define the open sets. The product topology uses open sets that are products of open sets from the individual spaces, while the box topology uses open sets that are unions of products of open sets from the individual spaces. This means that the product topology is generally finer than the box topology, and therefore has more open sets.

What are some applications of the product topology?

The product topology has many applications in mathematics, physics, and other fields. It is commonly used in algebraic topology to study product spaces of topological spaces, and in differential geometry to define tangent spaces and tangent bundles. It also has applications in the study of dynamical systems, functional analysis, and measure theory.

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