At what point does a cat become your's

The past few days I've heard a cat carrying on in the basement. I wasn't sure if he was trapped or if he just got in there somehow or what so I opened the door to the basement and he stopped meowing and just stood at the bottom of the stairs looking up at me so I brought him down a can of food on a paper plate but when I went down he ran away and hid somewhere but I noticed there was 2 little dishes of kibble and a big bucket of water down there already... So I set the plate down next to that and texted my neighbors in the other apartment in the house and asked if it was their's and they said yes and that I could have him cause his wife had an asthma attack because of him and they can't afford him and I told them I can't either but I can help out by bringing him food from time to time... My question is if I know he's in the basement that's partially mine and I bring him food from time to time does he become considered mine? I can't afford to take on the responsibility of another cat as far as vet stuff and so forth. And I'm not bringing an un-vetted cat into my apartment with my clean and healthy one anyway. I mean I have no idea if he has parasites or what. But if he does I don't want somebody saying I'm neglecting an untreated sick cat in my basement cause he's not mine...  The inside access to the basement is in my apartment but there's an outside door to it they used to put him down there.

If you're asking whether or not he'll legally be considered yours if you feed him, the answer is that it varies from municipality to municipality. In some places, if you feed a cat, you're legally the owner.  In others, if you get the cat veterinary care, it's legally yours, and in others, there are no such stipulations in place so it doesn't matter.  The bigger question is whether or not you want another cat. If neither you nor your neighbor can afford to care for this cat, the most humane thing to do is to try to find a home for the cat (asking friends, neighbors, co-workers, advertising on Facebook, putting up signs in supermarkets, etc) or take it to a shelter. Another option would be to take it to your local Humane Society with the intention of adopting it. The one in my area charges about 60 dollars for a cat, but that covers everything--spay/neuter, vaccinations, de-worming, and flea treatment--which would cost a few hundred dollars or more if you took the cat to the vet. In the mean time, I hope you and the neighbors will continue to feed him. I don't know what condition your basement is in, but it would be nice if someone provided him with a box with a blanket in it since it's getting cold now and a few cat toys. Poor little kitty.

Owning a pet comes with responsibility.  The most responsible thing, as MsAimee said, would be to find a home where he would be cared for.  Talk to your neighbors about options - do they know anyone who may want to adopt the cat?  You can help with posters, flyers, spreading the word, etc.  Ask your vet for help and all your friends, neighbors, family.

I also vote for providing him with the basics in the basement.  A litter box is a must or the place is going to get pretty rank, pretty soon.  A bed of some sort would be good, too.

Thank you for helping the poor creature.  Whatever you do is better than nothing!