Harbour is one of the many, many words which have a different spelling in Australian English than US English. We always include the "u"; the Yanks never do. But there are two fair dinkum Aussie examples where leaving out the "u" is actually the right approach.
The South Australian towns of Victor Harbor and Outer Harbor both have the u-free spelling, and always have had. Speculation suggests that a poorly-informed surveyor general back in the day made the mistake, and it has stuck ever since. Whatever the reason, it's yet another exception you'll need to learn if you're ever writing about South Australia. (It's worth going to Victor Harbor to try out the horse-drawn tram.)
Historical note: Variations between -or and -our in English began showing up in the 17th century, when there was a push to spell words of French origin with the -our ending, and those with Latin origins with an -or ending. Like most attempts to enforce spelling based on ancient etymology, this ended up confusing more people than it helped. In the US, standardisation eventually resulted in -or becoming the accepted suffix in virtually all circumstances, but UK (and hence Australian) English maintains a mixture of both variants.
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