If you just want the horse names, then possibly CSV is a little overkill - a simple list with one name per line is easy to read and write and needs no special processing:
Assuming you have your names in an array (just the ten you are interested in)
Write:
File.WriteAllLines(path, horses);
Read:
string[] horses = File.ReadAllLines(path);
If you have to use CSV, then see here:
Reading and Writing CSV Files in C#[
^] for a quick and simple version.