Being polite is something everyone should do for everyone else, not just for women.
I am no more polite to women, than I am to men, children, the elderly, whoever. I am polite to all strangers - I don't know them. If someone proves themselves to be a rude, wretched person then my politeness with them ends. And it also ends with friends and people I am close with. Not that I am rude to them, but politeness is very formal, and I'm not generally that formal with people I'm close to.
My main concern in being polite to women is having it thrown back in my face. I once held the door for a woman, who snidely said 'I can actually get that myself, thanks.' To me, that is the worst kind of person; one who throws a good deed at the doer. Suffice to say, I parted her with some less than polite words.
I think the term 'sexist' is too easy to throw around. If someone is polite to a woman, because of her sex, it that sexist? Its certainly making a distinction and a discrimination, albeit a positive one. However, in the modern world, packed full of strangers we must make decisions based on prejudgments all the time. We sort people by the obvious information available to us - sex, age, physical build, disability. We do not have time to learn about everyone's personalities and needs. If I see an old man, struggling with a bag, I may help him, because the elderly tend to have less muscle tone. If I am opening a door, which is quite heavy, and a woman is entering after me, I may hold open the door for her, because women tend to have less muscle tone than men. If I see a child alone, looking lost or distressed in a rough area, I will try to help them, because children are more vulnerable. If I see a woman lost and distressed in a rough area, I may also try to help her, because women are more vulnerable to attack than men. So if being polite and kind to women like that is sexist, then I'll rather happily be a sexist.
