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The phrase "are closely bound" is correct and usable in written English.
It can be used to describe a strong connection or relationship between two or more entities, ideas, or concepts.
Example: "The success of the project and the team's collaboration are closely bound, as effective communication is essential for achieving our goals."
Alternatives: "are tightly linked" or "are strongly connected."
Dictionary
are closely bound
adverb
In a close manner.
synonyms
Exact(16)
The two things are closely bound, interdependent.
Cycles of varying lengths are closely bound up with economic growth.
In societies where the main civil society networks are closely bound to the life of the churches, it is self-defeatingly foolish to bypass their systems.
"We know our lives are closely bound to the plant," said Hiroyoshi Hidaka, 56, who owns a health products store in Genkai.
"Through the euro we are closely bound together and the weakness of one affects us all," she said during the joint news conference.
The book has a dual focus, being the story of half brothers, Michel and Bruno, whose fates are closely bound up with the convulsive moral and spiritual sea changes of postwar France.
Similar(43)
His work, like his life, is closely bound up with contemporary Israeli life.
Because mental distress often has social links, this type of emergency room is closely bound to the world around it.
It might sound like a paradox but the fear of protest was closely bound up with the defence of liberty.
In theory, the classical and dance worlds ought to be closely bound, but they overlap less than you'd expect.
Northern Ireland's economy is closely bound to that of the rest of the United Kingdom.
More suggestions(22)
economy is closely bound
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are fully bound
are deeply bound
are closely bounded
are largely bound
are extensively bound
are closely bonded
are effectively bound
are closely associated with
are closely linked to
are closely linked together
are intimately bound
is intimately bound
are tightly bound
are closely intertwined
are closely entwined
are closely correlated
are closely packed
are strongly bound
are closely related
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Since I tried Ludwig back in 2017, I have been constantly using it in both editing and translation. Ever since, I suggest it to my translators at ProSciEditing.

Justyna Jupowicz-Kozak
CEO of Professional Science Editing for Scientists @ prosciediting.com