How many metric define on a nonempty set finite set or a countable set? Also on uncountable set.
Does this question make any sense?
I dig out in some books but I found nothing that answers my question. So I came to here for expecting some answers.
Asked
Active
Viewed 88 times
0
-
1There are clearly infinitely many. There is probably some simple set-theoretic argument to get the cardinality as well. A more interesting combinatorial question is how many metrics on a finite set $X$ there are if the distance function is constrained to take values in ${0,1,\cdots,N}$ for some nonnegative integer $N$. – MathematicsStudent1122 Oct 24 '20 at 04:36
-
On a finite or countable set the number of metrics is the cardinality of the continuum, whether or not we work up to isometry. This is not a very interesting result. More interesting would be to place some restriction on the values of the metric as MathematicsStudent1122 suggests. – Qiaochu Yuan Oct 24 '20 at 07:43
-
2See The number of continua by Lozier/Marty (1973), The number of metrizable spaces by Hodel (1983), On cardinality of sets of metrics generating metric spaces of prescribed properties by Šalát/Tóth/Zsilinszky (1992), (continued) – Dave L. Renfro Oct 24 '20 at 08:10
-
1Counting topologies by Kuba (2011), Counting metric spaces by Kuba (2011). – Dave L. Renfro Oct 24 '20 at 08:10
-
1If you are wanting to know how many different metric topologies there are (that is, how many different metrics you can define that not equivalent to each other), then the answer is $1$ for the finite case. For the countable space, there are still uncountably many metric topologies, depending on how many distrinct equivalence classes there are of Cauchy subsequences, from $\Bbb N$ at one extreme to $\Bbb Q$ at the other. – Paul Sinclair Oct 24 '20 at 15:14