What is the importance of Miss Brill's fur necklet in the story, Miss Brill?

Miss Brill's fur necklet, with its "dim little eyes," a nose "that wasn't at all firm," and a mouth that bites "its tail just by her left ear," assumes many human characteristics in the story. It is a friend to Miss Brill, who calls it her "little rogue," and whose eyes ask the question "What has been happening to me?"—a question that the woman is not able to ask of herself. The fur lives in a box, just as its owner lives in a "dark little room," and together they visit the park on Sunday afternoon.