Last July when construction started on a building across the road from my house, a small gray cat showed up in my yard. She would romp around the yard. 
After a few months, the little gray cat started to approach my husband and me. At first she would rub against our leg and then run away. 
One day she let my husband give her a pet. Shortly afterwards, there was a cold snap and we brought her in our home. We didn’t find her, she found us!
She is fiv positive which is not contagious among housecats and no special care is required.