Equivalence of d and p Metrics on X

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In summary, the conversation discusses proving that d and p are equivalent metrics on X. The attempt at a solution involves using the Lipschitz condition and considering the 'small' sets to determine the topology. It is suggested that showing a condition for d<=1 would suffice, even if it doesn't hold for all x,y but only for nearby ones.
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Homework Statement


Show that d and p are equivalent metrics on X where p=d(x,y)/(1+d(x,y))



Homework Equations


ive proved already that p is indeed a metric too (if d is a metric).


The Attempt at a Solution



I believe I am supposed to use the Lipschitz condition where there exits constants A and B st for all x,y,

Ap<=d<=Bp

but i think i can actually prove that one of these two constants cannot exist... and i using the wrong definitions? Thanks!
 
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Two metrics are equivalent if they induce the same topology. Isn't that the real definition? It is true that Ap<=d<=Bp shows that. But isn't it also true that for a metric topology what really determines the topology is the 'small' sets? If you can show there is a condition like that for say d<=1. That would also suffice, it doesn't have to hold for ALL x,y. Just for nearby ones.
 

1. What is the concept of "Equivalence of d and p Metrics on X"?

The concept of "Equivalence of d and p Metrics on X" refers to the idea that two different metrics, d and p, can be considered equivalent if they induce the same topology on the set X. In simpler terms, this means that two different ways of measuring distance or similarity between elements in a set can lead to the same overall understanding of the relationships within that set.

2. Why is the equivalence of d and p metrics important in scientific research?

The equivalence of d and p metrics is important in scientific research because it allows for different ways of measuring and analyzing data to be compared and understood as equivalent, even if they use different methods. This can lead to a more comprehensive understanding of complex systems and phenomena.

3. How can one determine if two metrics are equivalent on a specific set X?

To determine if two metrics, d and p, are equivalent on a specific set X, one can compare the topologies induced by each metric on that set. This can be done by examining the open sets generated by each metric and determining if they are the same. If the open sets are the same, the metrics are considered equivalent on that set.

4. Are there any examples of equivalent d and p metrics?

Yes, there are many examples of equivalent d and p metrics. One common example is the Euclidean distance metric and the Minkowski distance metric, which are both used to measure distance in different ways but lead to the same understanding of distances within a set. Another example is the cosine similarity metric and the Jaccard similarity metric, which are both used to measure similarity between vectors but can be considered equivalent in certain contexts.

5. How does the concept of "Equivalence of d and p Metrics on X" apply to real-world situations?

The concept of "Equivalence of d and p Metrics on X" can be applied to many real-world situations, such as data analysis, pattern recognition, and machine learning. In these fields, different metrics may be used to measure and analyze data, but the concept of equivalence allows for a more comprehensive understanding and comparison of results. This can lead to advancements in various industries, including healthcare, finance, and technology.

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