You might want to approach this question by examining one of the themes in the story and its development in the tale. The theme of isolation is an interesting one to look at, as Emily Grierson is part of this tightly-knit community where clearly everybody knows everybody else's business, and yet somehow, Miss Emily remains strangely isolated throughout her entire life. Note how at the beginning of the story Miss Emily is described as "a fallen monument" for which the men have a respectful affection for. She is likewise described as a "tradition, a duty, and a care; a sort of hereditary obligation upon the town." This is a rather strange statement to make when we consider how little Emily has been involved in the town and how defiantly she has kept herself to herself. In her childhood, it was her father who kept her isolated, but after his death, this is something that she continued herself, apart from her few shopping trips, jaunts with Homer Barron and the art classes that she gives.
However, although it appears Miss Emily couldn't care less about the town, she is clearly a topic of hot gossip. Again and again, the narrator talks of how her actions excite comment from all of the townspeople, especially concerning her relationship with Homer Barron:
So the next day we all said, "She will kill herself"; and we said it would be the best thing. When she had first begun to be seen with Homer Barron, we had said, "She will marry him." Then we said, "She will persuade him yet," because Homer himself had remarked--he liked men,and it was known that he drank with the younger me in the Elks' Club--that he was not a marrying man.
Throughout the story then, the townspeople hotly comment on her life and actions, revealing an interesting conflict. Even though Miss Emily regards herself as separate from the rest of the town, they regard her as one of their own.