Is it normal she wants to get with me so soon after breaking up?

One of my friends just broke up with her boyfriend. Without saying too much, it's not that they weren't good for each other, but they were victims to circumstance and it couldn't have continued. After that happened, she's been wanting to date me. She's been trying to get me interested in her with, though to her credit she's usually trying to be subtle. Not exactly doing it, it's too obvious both to me and our other friends, but she's trying. It's not that I wouldn't give her a chance, I actually want to. It's just that she broke up less than a week ago and she's probably having all kinds of emotions still. She also might just be using me as rebound and will give me the "better as friends" thing the moment she finds someone she likes more. I don't want to be in another relationship that's doomed from the start. I also think it's kind of strange for her to start looking for someone new so soon after her last, wonderful relationship ended. I guess it's good that she isn't getting hung up on it, especially because it was something that couldn't be helped, but I would think she would feel like she's betraying her last boyfriend by getting into a new relationship immediately after. My thoughts are to just give it a little time and go with it if she's still interested, but I might be losing an opportunity by not acting now. But if I do act now, I might just be setting myself up for disappointment. Is it normal for her to want to get into a new relationship right after her last one ended?