Understanding Metric Spaces: Subsets vs. Subspaces

[zolit]

Having some difficult with general concepts of metric spaces:

1) What is the difference between a subset and a subspace. let's say we have metric space X. and A is a set in that space. Is A necessarily a metric space itself?

2) Why is the metric of X ( d(x,y) for x,y belonging to X ) necessarily finite? Isn't the set of all real numbers a metric space, then how can you say that distance between any two numbers is finite?

Thanx!

 

[dextercioby]

Having some difficult with general concepts of metric spaces:

1) What is the difference between a subset and a subspace. let's say we have metric space X. and A is a set in that space. Is A necessarily a metric space itself?

Yes,a metric subspace A in X is a subset of the metric space X.

2) Why is the metric of X ( d(x,y) for x,y belonging to X ) necessarily finite? Isn't the set of all real numbers a metric space, then how can you say that distance between any two numbers is finite?

Thanx!

Why not?Give an argument in support of your question's logics...

Daniel.

 

[jcsd]

1) For a metric space X the metric is a function f:XxX-->R (that satisfies the correct conditons for a metric) , so a subspace is a subset A of the the set X together with the function g:AxA-->R such that f(x,y) = g(x,y) for all x,y in A and thus A forms a metric space in it's own right.

2) It's as the range of the metric function is the (nonnegative) real numbers and all real numbers are finite, the normal metric on the rela numbers is just d(x,y) = |x-y| clearly for any two real numbers x and y |x-y| is always finite.

 

[arildno]

Yes,a metric subspace A in X is a subset of the metric space X
Daniel.

Eeh, what he asked about, was whether an arbitrary subSET A was also a subSPACE. To that, the answer is simply no.
zolit:
It is important to remember that a "space" is a "set", where we have defined that there exist some (element) addition operation and scalar multiplication operation.
Furthermore, in order to be a "space" certain properties about our "set" must hold (closure properties and so on)

 

[jcsd]

arildno, thoguh for any metric space any subset forms a subspace with the correct metric (the concept of a metric space is more primitive than the concept of a vector space with a metric)

 

[dextercioby]

Arlidno,reread the first paragraph of his post.U'll find two questions.I chose to answer the second,simply because it was easier to give an answer to... Besides,i knew someone more knowledgeable than me would come up with a more detailed answer than i could have offered.

Daniel.

 

[arildno]

arildno, thoguh for any metric space any subset forms a subspace with the correct metric (the concept of a metric space is more primitive than the concept of a vector space with a metric)

The shame of it..(goes and hides in a bucket)

 

[jcsd]

The shame of it..(goes and hides in a bucket)

'Tis an easy mistake to make, if your training is biased towards physics the concept of a metric only seems to pop up with relation to vector spaces in physics, except in the more mathematics based areas.