How you spell this word depends on where you live in the English-speaking world.
Well-being as a hyphenated word is the preferred form in the United States and Canada.
Wellbeing as one word with no hyphens is the standard spelling in the United Kingdom, Australia and other English-speaking countries outside of North America.
The word never should be spelled as two words without a hyphen; well being doesn’t make sense as it isn’t two nouns stuck together to form a new object (and hence a new word), as is the case with ice cream.
Be forewarned that as the less preferred form of each word often appears no matter where you live, sometimes spell checks won’t catch the error.
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My name is Rob Bignell. I’m an affordable, professional editor who runs Inventing Reality Editing Service, which meets the manuscript needs of writers both new and published. I also offer a variety of self-publishing services. During the past decade, I’ve helped more than 300 novelists and nonfiction authors obtain their publishing dreams at reasonable prices. I’m also the author of the 7 Minutes a Day… writing guidebooks, four nonfiction hiking guidebook series, and the literary novel Windmill. Several of my short stories in the literary and science fiction genres also have been published.