Statt dessen leben sie so lange alleine, wie sie selbst für sich sorgen können.

I thought stattdessen was one word?

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According to my quick research you are correct, and since the orthography reformation it is now spelt as a single word.

However this is only true if the meaning is “instead”. There are contexts in which two separate words are still correct:

Er entschied sich, Markus in die Geschehnisse einzuweihen, statt dessen Vertrauen zu verlieren.
He decided to let Markus in on what was happening instead of losing his trust.

Here, “statt” means “instead of”, and “dessen” corresponds to “his”.
In this sentence, the construction is “statt zu …” (or “anstatt zu …”), and “dessen Vertrauen” is a separate unit, which is why it’s separate words.

That being said, there are millions of google hits where it’s written separately. Even natives are obviously not always sure, and certainly noone will blame you for writing it as two words.

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