I did see a spider running very quickly across the floor from near the area I took my sweater off the last time. It was a rather large one and when I look at pictures of a recluse spider I'd say that's what it was. Sadly, I squished it.
It might have been a brown recluse, but then again it might not have. It's extremely difficult to accurately identify spiders without a background as an arachnologist. Even the experts have a hard time identifying a brown recluse, even with a good specimen and time to examine it.
If you think you know the spider species, by all means tell the doctor. However, a good doc knows not to put too much stock in guessing, so he or she should still consider all the possibilities. Assuming it's a spider bite when it's really something else can be dangerous.
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Any opinions expressed here are for educational purposes only and are not intended for diagnosis.