The boy recognizes the courage of the old man becomes fond of him for his spirit; however, this admiration would not be as uncommon in the culture of the boy. Then, through their love of fishing and of baseball, Santiago and the boy grow closer.
It is sad when a person becomes old and has no family, or has a family that has abandoned him. Most young people are not interested in helping an old man. Therefore, the relationship between the old man and the boy is an unusual one.
The old man and the boy counteract to contribute to the overall theme of the story. Their relationship is based on friendship and they both rely on each other for warmth and love. The old man has no one to rely on or to take care of him and this is where the boy comes in. Although the relationship is solely based on friendship, the boy voluntarily looks after the old man and the old man does the same.
I have to agree with post #2 in that a brief response probably won't suffice here. We can say that the old man's relationship to the boy is one of friendship, but that doesn't get at the importance or the symnbolism of the relationship the two share.
The old man and the boy are opposites in many ways, yet they believe in one another. In way we can also say that each has what the other does not, one possessing youth and a future and the other possessing only a past.
You need to post these on the question and answer part of the website. This kind of question requires an in-depth answer from someone who has knowledge of the specific work you are asking about. Discussions are meant for shorter responses, usually based on someone's personal thoughts and opinions.