Here is one my example on how to create basic sitemap using serialization XML should look like:
<?xml version="1.0" encoding="utf-8"?>
<urlset xmlns:xsi="http://www.w3.org/2001/XMLSchema-instance" xmlns:xhtml="http://www.w3.org/1999/xhtml" xsi:schemaLocation="http://www.sitemaps.org/schemas/sitemap/0.9 http://www.sitemaps.org/schemas/sitemap/0.9/sitemap.xsd" xmlns="http://www.sitemaps.org/schemas/sitemap/0.9">
  <url>
    <loc>http://www.example.com/</loc>
    <lastmod>2005-01-01</lastmod>
    <changefreq>monthly</changefreq>
    <priority>0.8</priority>
  </url>
</urlset>
The model:
using System.Collections.Generic;
using System.Xml.Schema;
using System.Xml.Serialization;

namespace SiteMapXmlCreation
{
    public class Model
    {
        [XmlRoot("urlset")]
        public class Urlset
        {
          [XmlAttribute("schemaLocation", Namespace = XmlSchema.InstanceNamespace)]
          public string SchemaLocation { get; set; }
            [XmlElement(ElementName = "url")] public List<url> Url { get; set; }
        }

        public class url
        {
            [XmlElement(ElementName = "loc")] public string Loc { get; set; }

            [XmlElement(ElementName = "lastmod")] public string Lastmod { get; set; }

            [XmlElement(ElementName = "changefreq")] public string Changefreq { get; set; }

            [XmlElement(ElementName = "priority")] public string Priority { get; set; }
        }
    }
}
The Program:
using System.Collections.Generic;
using System.IO;
using System.Xml.Serialization;

namespace SiteMapXmlCreation
{
    class Program
    {
        static void Main(string[] args)
        {
            Model.Urlset serializedUrlset = new Model.Urlset
            {
                Url = new List<Model.url> {
                    new() {
                        Changefreq = "monthly"
                        , Lastmod = "2005-01-01"
                        , Loc = "http://www.example.com/"
                        , Priority = "0.8"
                    }
                }
                , SchemaLocation = "http://www.sitemaps.org/schemas/sitemap/0.9 http://www.sitemaps.org/schemas/sitemap/0.9/sitemap.xsd"
            };

            TextWriter txtWriter = new StreamWriter(Path.ChangeExtension(System.Reflection.Assembly.GetEntryAssembly().Location, ".xml"));

            XmlSerializerNamespaces ns = new XmlSerializerNamespaces();
            ns.Add(string.Empty, "http://www.sitemaps.org/schemas/sitemap/0.9");
            ns.Add("xsi", "http://www.w3.org/2001/XMLSchema-instance");
            ns.Add("xhtml", "http://www.w3.org/1999/xhtml");

            XmlSerializer xmlSerializer = new XmlSerializer(typeof(Model.Urlset), "http://www.sitemaps.org/schemas/sitemap/0.9");
            xmlSerializer.Serialize(txtWriter, serializedUrlset, ns);

            txtWriter.Close();
        }
    }
}
Example download from here.