How Does the Maximum Metric Define Topology on the Product of Two Metric Spaces?

In summary, the conversation discusses the product of two metric spaces and defines the metric for this product as the maximum of the metrics for each space. The conversation also mentions proving that this metric is indeed a metric and discussing the basis for both the metric topology and the product topology on X x Y.
  • #1
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Homework Statement [/b]
Let (X, dX ) and (Y , dY ) be metric spaces. The product of X and Y (written X × Y ) is the set of pairs {(x, y) : x ∈ X, y ∈ Y } with the metric:
d((x1 , y1 ), (x2 , y2 )) = max {dX (x1 , x2 ), dY (y1 , y2 )}
1)How to prove that d is a metric on X × Y?
2)Prove that d induces the product topology on X × Y.
 
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  • #2
To prove it is a metric, just go through the axioms of being a metric, taking into account that dX and dY are metrics on X and Y respectively (and satisfy all the axioms).

Notice that X x Y is a finite product. What is a basis for the metric topology that d induces? What is a basis for the product topology on X x Y?
 

FAQ: How Does the Maximum Metric Define Topology on the Product of Two Metric Spaces?

1. What is a product of two metric spaces?

A product of two metric spaces is a mathematical construction that combines two metric spaces into a new metric space. It is denoted by X x Y, where X and Y are the two metric spaces. The elements of the new metric space are ordered pairs (x,y) where x is an element of X and y is an element of Y. The distance between two elements (x1,y1) and (x2,y2) is defined as the maximum of the distances between x1 and x2 in X and between y1 and y2 in Y.

2. What is the purpose of a product of two metric spaces?

The purpose of a product of two metric spaces is to provide a way to combine two metric spaces into a single space while preserving certain properties. This allows for the study of the combined space as well as the individual spaces separately. It also allows for the construction of new spaces with desired properties.

3. What are some examples of product of two metric spaces?

Some examples of product of two metric spaces include the Euclidean plane (R2), which is the product of two real lines (R x R), and the product of two discrete metric spaces, which results in a discrete metric space with elements being ordered pairs. Other examples include the product of a metric space with itself, known as the square of the space, and the product of a finite number of metric spaces.

4. How is the product of two metric spaces different from the direct sum of the spaces?

The product of two metric spaces is different from the direct sum in that it considers the maximum of the distances between the elements of the two spaces, while the direct sum considers the sum of the distances. This results in different topologies for the two constructions.

5. What are some applications of the product of two metric spaces?

The product of two metric spaces has applications in various fields, such as functional analysis, topology, and geometry. It is also used in the study of product measures in probability theory and in the construction of product spaces in statistics. Additionally, the product of two metric spaces is used in computer science for data structures and algorithms, such as multidimensional arrays and graphs.

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