What is the significance of strong equivalence in topological space metrics?

In summary, a metric dp on the topological space X×Y is defined as dp((x,y),(x1,y1))=((dX(x,y))p+(dY(x1,y1))p)1/p. As p approaches infinity, d∞ is equal to the maximum of dX(x,y) and dY(x1,y1). The function max(A,B) returns the larger of A and B. Two metrics d and d* on a metric space Z are considered strongly equivalent if there exist positive constants M and m such that m d(x,y) < d*(x,y) < M d(x,y) for all x and y in Z, meaning that the two metrics generate the same topology on Z
  • #1
ForMyThunder
149
0
A metric dp on the topological space X×Y, with dX(x,y) and dY(x1,y1) being metrics on X and Y respectively, is defined as

dp((x,y),(x1,y1))=((dX(x,y))p+(dY(x1,y1))p)1/p

What does each dp((x,y),(x1,y1)) mean (geometrically or visually)?

as p[tex]\rightarrow[/tex][tex]\infty[/tex],

d[tex]\infty[/tex]=max((dX(x,y),dY(x1,y1)).

What is the meaning of "max(A,B)"?

Each of the dp((x,y),(x1,y1)) are strongly equivalent; what does this mean geometically?
 
Physics news on Phys.org
  • #2


You probably want p>=1 for d_p to actually be a metric. Anyway, have you tried putting X=Y=R? This should be revealing, e.g. if p=1 you get the "taxicab metric" and if p=2 you get the usual Euclidean metric. For other p>1, you may want to sketch the unit balls of (R^2, d_p) to get a feel for what the metric d_p does.

max(A,B) means what you would expect it to be; namely, max(A,B) is A if A>=B and B otherwise.

Finally, two metrics d and d* on a metric space Z are said to be strongly equivalent if you can find positive constants M and m such that the string of inequalities
m d(x,y) < d*(x,y) < M d(x,y)
holds for all x and y in Z. This essentially means that inside each d-ball you can find a d*-ball and vice versa. In particular, this implies that the metrics d and d* generate the same topology on Z.
 
Last edited:
  • #3


Thanks. I've been worrying about what they were fr a while. It seems kinda obvious now.

Thanks again!
 

1. What is a topological space metric?

A topological space metric is a mathematical concept used in geometry to measure the distance between two points in a topological space. It is a function that assigns a non-negative real number to every pair of points in the space, satisfying certain axioms.

2. What is the difference between a topological space metric and a metric space?

A topological space metric is a specific type of metric defined on a topological space, while a metric space is a general mathematical concept that includes topological space metrics as a special case. In other words, all topological space metrics are metric spaces, but not all metric spaces are topological space metrics.

3. What are some common examples of topological space metrics?

Some common examples of topological space metrics include the Euclidean metric, which is used to measure distances in Euclidean space, and the taxicab metric, which is used to measure distances in a city grid. Other examples include the discrete metric, the maximum metric, and the Manhattan metric.

4. Can a topological space have more than one metric?

Yes, a topological space can have multiple metrics. In fact, different metrics on the same topological space can result in different topologies, which can lead to different notions of convergence and continuity. This is one of the key differences between topological space metrics and other types of metrics.

5. What are some applications of topological space metrics?

Topological space metrics have various applications in mathematics, physics, and computer science. In mathematics, they are used to study the properties of spaces and to define continuous functions. In physics, they are used to model physical systems and to describe the behavior of particles. In computer science, they are used in algorithms for data clustering, image processing, and machine learning.

Similar threads

  • Precalculus Mathematics Homework Help
Replies
17
Views
908
  • Set Theory, Logic, Probability, Statistics
Replies
1
Views
707
  • Topology and Analysis
Replies
9
Views
2K
  • Topology and Analysis
Replies
10
Views
2K
  • Calculus and Beyond Homework Help
Replies
3
Views
110
Replies
4
Views
996
  • Calculus and Beyond Homework Help
Replies
2
Views
709
Replies
35
Views
2K
  • Topology and Analysis
Replies
4
Views
1K
  • Set Theory, Logic, Probability, Statistics
Replies
19
Views
1K
Back
Top