It is the initial paragraph of this excellent short story that we can call the exposition, as it introduces us to the central character and the situation that initiates the conflict. Consider how this is achieved in an incredibly concise manner in the first opening paragraph:
Knowing that Mrs. Mallard was afflicted with a heart trouble, great care was taken to break to her as gently as possible the news of her husband's death.
We are thus introduced to the central character, Mrs. Mallard, whose husband has just apparently died. We are also told, in a detail that is easily overlooked but has massive importance for the ending of the story, that she suffers from some kind of unspecified heart trouble. Clearly, the beginning of this story leads us to fear that this condition will result in the shock of the news of her husband's death being too much for Mrs. Mallard, and that she too will die. Of course, the first sentence of this story can be usefully compared with the last sentence, giving the story its grim irony:
When the doctors came they said she had died of heart disease--of joy that kills.
Note how this detail that is provided for us in the opening sentence is referred to again in the final sentence, as the doctors conclude that it was the joy of seeing her husband again that killed Mrs. Mallard, when we as readers know it was actually the thought of returning to the state of marriage after she had tasted blissful freedom which killed her. Thus we can see the importance of the opening paragraph both as an exposition but also in terms of helping to set up the irony of this excellent short story.