Friend Problem

I have a lot of issues making and keeping friends, so this particular problem bugs me a lot.
There is a girl I met at college when I was very lonely who was also lonely and very sad, and who I didn't want to be more than casual acquaintances with but who could probably use a friend. So at the time, I would hang out with her after our class, and talk about literature (the only thing besides loneliness and social awkwardness we have in common).
This girl is rather close minded, and I didn't think this was a problem because I figured if we talked for a while maybe she'd decide to question some of her more unfounded convictions...
Anyways, I guess my problem is I feel more mature than her, so I don't feel much of a connection and she keeps contacting me on social media but I don't want to encourage her too much.
I feel so guilty, because I do want to have some friends, and feel as if I don't have any at the moment, but I think being friends with someone because you pity them is unfair. I suspect that a person I considered to be my best friend stayed with me long after she lost interest because she pitied me. Should I keep up the acquaintance to any extent? I don't wish to be cruel, but we did only see each other after class for a couple of months.
(This whole affair makes me feel worse about myself...and makes me wonder how I can judge anyone. If I was worthy of friends, wouldn't I have some...?)

if you have no common interests and only saw each other after class for a couple of months than I wouldn't even call that a friendship. that is aquaintences. that doesn't mean you have to defriend her on social media. lord knows, I would have no friends left on facebook if I defriended everybody I didn't have a deep and meaningful relationship with...lol...nearly everybody is casual. but that doesn't mean you have to hang out with her. people may post on my wall or poke me.....sometimes I send a smily icon back just to acknowledge them, or a thanks or a like, but not always. if you get invited somewhere, you just respectfully decline.

I agree with kalope, completely.