It may be their way of saying that they still have concern for you, and it is their way of connecting to you. .
Reaching out from time to time
If your ex reaches out to you on special holidays like Christmas, it could be her/his perfect excuse to get connected with you again.
Showing signs of emptiness/loneliness
When your ex constantly lets the whole world know that they are gloomy on social media, it could be a sign that they are longing for your attention.
Trying to patch things up
They may constantly state how much has changed and how much better a person they are right now
Available to help you out
When your ex is always available whenever you need help, this is a good indication that they want you back in their life
Still single after a long time
If a long time has passed since your breakup and your ex has not gone out dating ever since, probably they still hope of getting back with you.
Becoming less social
Not engaging with the world could be a sign of depression. After the breakup, your ex may be depressed because the person that gives meaning to their life is gone.