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I am trying to understand what metric space is (I need it to understand what Minkowski space is, as one of the key to understand why Twins paradox is not a paradox (if it is not indeed)).

I understood that metric space is described by a set and a metric.

First confusion

A set, as I understand is just a bunch of numbers, that space consists of. Here I already have doubts. Is it some specifically described bunch of numbers (by a function, for example, or manually), or it meant a group of numbers from the hierarchy, like Natural, Rational, Complex, etc.?

Second confusion

A metric, as I understand, is just a function, that defines how to calculate the distance between … and here I also have confusion.

Distance between points (which can be vectors entities, i.e. be described by two or more numbers) made up from numbers from the Set, or “distance” between directly two (or more?) numbers from the Set.

The second one actually means some fundamental stuff, i.e. it should describe, what is the distance between numbers, like between 5 and 3, so it rule can be, that it wouldn’t be 2.

I always see in literature, that metric is written as

$$d(x,y)$$ $$d(x,x)=0$$

What does x, and y mean there?

Third confusion

Probably, more general, a metric $d$ of some set $M$ is described as

$$d: M \times M -> R$$

What does it mean? Does $\times$ mean cross product? Why $M$ is twice?

Stdugnd4ikbd
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  • Keep in mind that Minkowski space is not a metric space. – Moishe Kohan Feb 16 '23 at 22:30
  • @MoisheKohan, well, yes, maybe I a bit confused, but, as written on Wikipedia, it’s kinda metric space. At least, I think, it will be useful to clearly understand what metric space is, to understand what pseudo-Euclidian space is – Stdugnd4ikbd Feb 17 '23 at 09:34

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A set is just a collection of objects. The objects are not necessarily numbers; variables, geometric shapes, or even other sets would do just as fine. For instance, $A:=\{\text {apple, \{banana, orange\}}\}$ is a set.

A metric space is an ordered pair $(M,d)$ where $M$ is a set and $d$ is a metric on $M$. A metric on $M$ is a function from $M\times M$ to $\mathbb{R}$ that satisfies certain properties here. Also, $\times$ is the Cartesian product.

In plain words, a metric takes in two elements from the set $M$ and maps them to a real number that represents a notion of "distance" between those two elements. So the notation $d(x,y)$ represents a distance between two objects $x,y$ that live in $M$. The linked wiki page also has common examples of metrics.

Golden_Ratio
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  • Thanks, actually I read that article before asking. Now I understand, what that cross means. – Stdugnd4ikbd Feb 17 '23 at 10:04
  • Although, still, have a bit confusion: if there is a set $M$ with entities of different type, for example: M = [46(number), 1,2,3], how do I write metric function, in particular it’s arguments? – Stdugnd4ikbd Feb 17 '23 at 11:35
  • @Stdugnd4ikbd You just assign a distance for every pair of elements from the set $M$ in a way that obeys the properties of a metric, whether or not the elements are the same "type" of object, e.g. $d(46, (1,2,3))=1$. If $M$ is finite, that would mean specifying $M\choose 2$ distances. "Type" of object doesn't matter; for instance, we may measure the distance between a point and a plane. – Golden_Ratio Feb 17 '23 at 13:52
  • Okay, but how do You write it as a formula? In Your example, You have already passed, arguments (46 and (1,2,3)), and did not specify the equation of the function, just wrote the result. In case of arguments, probably it can be just x and y, undependably on type. But how in function's equation You will know, what type is variable, if it can various? – Stdugnd4ikbd Feb 17 '23 at 14:20
  • For example in some programming languages, in C++, in particular, type of variables should be specified in function (as well as return value of function): int func(int a, Array b). In other languages, in function's body programmer can determine variable type using conditional statements: if(typeof a == 'integer')dosomething; – Stdugnd4ikbd Feb 17 '23 at 14:27
  • I probably rewrite my question a bit. Although, no, I will ask another question – Stdugnd4ikbd Feb 17 '23 at 14:28
  • @Stdugnd4ikbd A function does not need to be written as a compact formula. Please check out the definition of a function. A function is just a map that specifies a value in the codomain for every element in its domain. Nothing more. For your example, $M$ just had two elements so $M\times M$ has four elements. So a function on $M\times M$ needs to assign values to four elements. However, it is enough to specify the value of $d(46,(1,2,3))$ because properties of a metric pin down the other values (check the wiki page for properties). – Golden_Ratio Feb 17 '23 at 15:47
  • @Stdugnd4ikbd In summary, I would suggest thinking about each element in a set as a generic object. "Type" may be important in a programming context, but not in this particular context. – Golden_Ratio Feb 17 '23 at 17:50