# Metric Space confusion

I have a simple question about the notation. I want to be more correct with notation, I don't understand exactly what the notation is saying.

In regards to a Metric space

A metric space is an ordered pair (M,d) where M is a set and d is a metric on M, i.e., a function

${{\bf{d: M \times M}}}$ (the syntax here)

such that for any x,y,z $\in \bf{M}$ (and the e looking symbol)

Does the times symbol in the first part indicate an M by M matrix, how would this be read. And the e symbol which I don't know the name of, I believe is akin to saying all numbers in, or every set in the space, or something roughly like that. So, how would I read the whole thing given the mathematical statement as written above if there was no explanation surrounding it? I have a lot of trouble reading pure math books sometimes because I don't understand a lot of the simple notation, even though I've seen the material before. I don't necessarily know how to represent the statement mathematically, for instance if I wanted to say something like a function that takes all real numbers as arguments, or considering all real numbers.

pwsnafu
I have a simple question about the notation. I want to be more correct with notation, I don't understand exactly what the notation is saying.
First thing I want to say, is congrats. Notation is something a lot of students underestimate, so I applaud you on this.

In regards to a Metric space

A metric space is an ordered pair (M,d) where M is a set and d is a metric on M, i.e., a function

${{\bf{d: M \times M}}}$ (the syntax here)

such that for any x,y,z $\in \bf{M}$ (and the e looking symbol)

Does the times symbol in the first part indicate an M by M matrix, how would this be read. And the e symbol which I don't know the name of, I believe is akin to saying all numbers in, or every set in the space, or something roughly like that. So, how would I read the whole thing given the mathematical statement as written above if there was no explanation surrounding it? I have a lot of trouble reading pure math books sometimes because I don't understand a lot of the simple notation, even though I've seen the material before. I don't necessarily know how to represent the statement mathematically, for instance if I wanted to say something like a function that takes all real numbers as arguments, or considering all real numbers.
The times symbol, when used with sets, is called the Cartesian product. If you don't know set theory, then treat it as "d is a function of two variables, both chosen from M"

The e symbol is the "element of" symbol. ##x \in X## means that X is a set, and x is a member (element) of that set. For example, if X is the set denoting the integers, then x is an integer.

My recommendation to you is to first read a book on basic set theory before dealing with metric spaces. A book like "How to prove it" by Velleman should suit your purposes (although it contains much more material than is necessary).

Thank you for the clear explanation, it is definitely helpful. My concern with the importance of notation stems more from me being easily confused by ambiguous notation and unclear notation than anything else.

Also, I looked up the set theory book, and it wasn't very expensive to get a used older version of it from 1994, so i went ahead and bought it. Thank you! I was having trouble finding set theory material to study because it seems like there is a large amount of set theory out there, and since it is used in basically every branch of mathematics, it's everywhere.