Thread: Set without a metric

Hello everyone,

I seem to have trouble with getting my head around a metric space.
I read the definitition of a metric which states that it is a function describing the distance between two points in a set, that is the shortest distance.
Then a metric space is a set where the notion of distance between elements of the set is defined.
I do not get how the distance between two points in a set can not be defined, so I was looking for an example of a non-metric set.
Unfortunately i could not find any, could someone shed some light on this issue for me?

Thanks!

Let X= {a, b, c, d}. (a, b, c, and d do NOT represent point- they are, literally, the four letters.) That's a set and there is no metric ("distance") defined on it. (You may be thinking of "sets of points" in a Euclidean space. That is not necessarily the case.)

Another example: P is the set of all polynomials in the variable x. We do not define the "distance between two polynomials".

Originally Posted by Mkkl

I seem to have trouble with getting my head around a metric space.
I read the definitition of a metric which states that it is a function describing the distance between two points in a set, that is the shortest distance. Then a metric space is a set where the notion of distance between elements of the set is defined.
I do not get how the distance between two points in a set can not be defined, so I was looking for an example of a non-metric set.

Surely you know that there are many many abstract sets that have no ordinary metric defined on them.
The set of all continuous functions on $\displaystyle [0,1]$ for example. We can certainly define a metric for that set, there are several different ones. But really no one of them is completely natural.

Here is a real world example. In a large city the streets may form a strict grid or lattice. Buildings make it impossible to get from on corner to the opposite along a straight line (the diagonal).

Say we are at point $\displaystyle (2,2)$ and need to be at $\displaystyle (6,5)$.
Normally we would say that is a distance of $\displaystyle \sqrt{(6-2)^2+(5-2)^2}=5$ units. That is called the Euclidean metric.
But the buildings mean we can only travel along the streets. Thus we must travel seven blocks or units not five.

In a lattice the distance from point $\displaystyle P: (a,b)$ to point $\displaystyle Q: (c,d)$ is defined as $\displaystyle D(P,Q)=|a-c|+|b-d|$ and is known as the city block metric.

Hey there!

Thanks for your reactions. I thought there may have been somewhat less trivial non-metric sets, but I may be thinking to difficult here.
The example of the city block metric was most helpful for me.

Thank you for your time!

Originally Posted by Mkkl

Hello everyone,

I seem to have trouble with getting my head around a metric space.
I read the definitition of a metric which states that it is a function describing the distance between two points in a set, that is the shortest distance.
Then a metric space is a set where the notion of distance between elements of the set is defined.
I do not get how the distance between two points in a set can not be defined, so I was looking for an example of a non-metric set.
Unfortunately i could not find any, could someone shed some light on this issue for me?

Thanks!

The discussion appears to be having ups and downs. If the set is described as the distance between tow elements of the set then where is the problem. When we refer to the distance between two points it is supposed to be the shortest distance. With this much one can definitely proceed to answer the questions raised.