This is ONE OF 3 responses to Vol 8 John 19 ("Friends and Lovers")...
You have defined the discussion to oblivion by your use of the word "close". If we instead rephrase the question to can ex-lovers be friends then the discussion can become alive again.
I feel that friends can, very roughly, be put into three categories, casual, good, and close, and that most people recognize these rough categories. Often, very often, in discussion, they don't clearly define the category they are discussing and this is the major reason for the disagreement.
Casual friends, usually neighbors, coworkers, and fellow club members are trusting, helpful, usually have some common interests, and are friendly. You can trust them to give you a fair break. There is some restraint between them because they are not that well acquainted.
Good friends are casual friends with the addition of deeper trust, stronger feeling of helpfulness and responsibility, better knowledge of each others feelings and less restraint in conversation.
Close friends are good friends with the addition of the deepest feelings of trust, knowledge of each other feelings, the least restraint in conversation. You can ask big favors of them and will respond with strong actions and both have many interests in common. Close friends would be lovers unless (a) one was gay, (b) both disliked sex, or (c) both believed strongly that "good people" must exclude sex, even thoughts of sex with anyone but their wife/husband and so strongly that the thought of it wouldn't even enter their minds.
If you accept these definitions, then it becomes obvious that ex-lovers COULD become casual or good friend after they break up but not close friends because they would have to share feelings and emotions so closely and trustingly that they would not have broken up. The breakup indicates that the feelings and emotions that was shared is now not shared.
Do you want to stick tightly to the question Can ex-lovers become close friends or open it up to Can ex-lovers be friends? Or do we need to define "friends" more closely.
My personal experience with this situation was that demanding or even expecting that your spouse totally renounce seeing old friends of the opposite sex is the first step toward disaster in a marriage because it basically demonstrates first a lack of faith in the spouse and secondly shows the childish emotion of jealousy. And it contributed to a divorce in my particular case.
[Editor's note: I continued my arguments in my response to Larry's card; See Vol 9 John 18]