I saw a road sign that said: this sidewalk to be closed on or about. I am not sure what this means.
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It sounds like someone forgot to finish the sign... A sign that says that a sidewalk is to be closed "on or about" is probably meant to have someone write in the date "on or about" which the sidewalk is to be closed.
In normal usage, the phrase "on or about" is followed by a date and has the same meaning as "approximately." If a sign says that a sidewalk will be closed on or about January 5, 2014, I wouldn't be surprised if it were closed anytime between a few days before to a few days after January 5, 2014.
The phrase has a more precise legal definition when used in legal documents.