In City of Ashes, the theme of betrayal is explored in a number of ways. First, the main characters are betrayed by their friends and allies. For example, Jace is betrayed by Sebastian, who he thought was his brother. Clary is betrayed by Simon, who she thought was her best friend. And Alec is betrayed by Magnus Bane, who he thought was his lover.
Second, the main characters are betrayed by the institutions they trust. For example, the Clave is betrayed by the Circle, which is a group of rogue Shadowhunters. And the Shadowhunters are betrayed by the Greater Demons, who they thought were their allies.
Third, the main characters are betrayed by their own desires. For example, Clary is betrayed by her desire to be like her mother, Jocelyn. And Jace is betrayed by his desire to be like his father, Michael Wayland.
The theme of betrayal is important in City of Ashes because it shows how the main characters are able to overcome the challenges they face. By confronting their betrayers and learning from their experiences, the main characters are able to become stronger and more resilient.