How are Scrooge and fan his sister very close?

In several versions of A Christmas Carol, Scrooge is seen talking to his clearly smaller sister "Fan" and has, on several accounts, referred to her as "little Fan".

It is also explained that Scrooge's mother died giving birth to him.

Based on these two facts, it must be assumed that Scrooge's presumably, if not obviously, younger sister was the result of a second marriage.

Thus, Fan is likely Scrooge's half-sister.

an excerpt from the original text confirms your suspicions.

" He was not reading now, but walking up and down despairingly. Scrooge looked at the Ghost, and with a mournful shaking of his head, glanced anxiously towards the door.

It opened; and a little girl, much younger than the boy, came darting in, and putting her arms about his neck, and often kissing him, addressed him as her "Dear, dear brother."

"I have come to bring you home, dear brother!" said the child, clapping her tiny hands, and bending down to laugh. "To bring you home, home, home!"