I think it is safe to say that most people believe that men holding doors for women represents some kind of chivalry or gentlemanliness.
Men feel gracious and powerful when they do it, while women tend to feel flattered and or deserving.
I on the other hand have things to argue about this social reflex.
People should hold doors open for people.

This door-holding business imitates a sense of true aid. It makes it seem like the man is willing to help women in lots of different aspects of life. From standing on the car-side of the sidewalk to paying the bill to throwing his own laundry in the laundry bin, he may feel he is the absolute "man" and his lady would be nothing without those helpful little hands. (Where are the men when women need help with domestic labour, doing his laundry, making meals for the both of them?)

How dare a man who is holding two T.V.s in each hand while balancing a record player on his head and wobbling with a DVD player in between his legs let a woman open the door for him? *spits* And how dare he not drop everything to make sure her empty-handed self can glide through the door? Men tend to place these gallantries above the practical reality of the situation. The door-opening and other small services are intended for people who are generally incapacitated, burdened or generally unable in some way to do that thing for themselves. Wouldn’t this mean then that if a person was carrying something while approaching a doorway that anyone near should rush ahead a bit to open the door - regardless of gender? In this sense, it seems obvious that these tiny tasks (that even sometimes woo women) are actually just actions that reinstate their inferiority and place men as the only mover for the unmoved and the only ones willing to literally step forward and do something.

Don't get me wrong, I appreciate when people hold the door open for me and never fail to say "Thank You". What I cannot handle is when I do hold the door open for a guy and he stops before he crosses the threshold so that I can walk away and he can open the door for himself to walk through. Another situation is when I stop walking to hold a door open and the man in question puts his hand on the door while he walks through (sometimes even using a whole forearm), as if I need help doing it. Then he says "Thanks", in a way that I take as a pitiful "Thanks for trying". Or maybe because he is insulted that I'd dare to doubt his ability to do it for himself - or his manhood in not doing it for me,. Just let me hold the fucking door open for you. I'll be fine, I know I will.