We in Oklahoma got clobbered this Christmas eve with snow. We are not used to having 8"-12" of snow ever, let alone on Christmas. The weather threw everyone's plans into disarray and many were caught in snow drifts trying to get home.
I say this not to minimize the difficulty we all had, but to make the point that as a business owner you need to inform your (prospective) customers about what is going on.
If you were forced to close early, the least you should do is to put a sign in the door, then change the answering machine for the store phone - "Due to the weather we are closed until Friday". You should also take a couple minutes to do the same for your website main page. Both of these could be done from your own home, without requiring you to drive into the store. (You do have your phone answering machine codes and website logins at hand, don't you?)
My wife and I braved the roads on Christmas day to go see a movie. When we arrived at the theater, it was closed. We had actually called the theater beforehand, and it listed the movie times and made no mention it was closed. If my wife and I had been the only ones caught, you might have argued we were just crazy to go. But during our time there at the theater, at least 10 other cars pulled in the lot, each driving up to the door trying to go to the movies. It is not outside the realm of possibility that someone might try to go see a movie on Christmas.
Don't compound the bad weather by not considering your customers!