独立(dú lì) - independent, independence; to stand alone in Mandarin

独立 (dú lì) stretches from the political to the personal. A country 获得独立, a young adult 学会独立, a room can be 独立的 in the sense of self-contained, and two questions can be considered 独立 of each other.

It is flexible about its grammatical job. As an adjective it takes 很 like any other (他很独立). As an adverb it modifies a verb directly (独立完成这项工作). As a noun it fills the object slot after verbs such as 获得 or 宣布.

The trap is the English preposition. "Independent of something" needs 于 in Mandarin, giving the written 独立于, and that pattern is formal. In speech you would rewrite it: 跟那件事没有关系 says the same thing more naturally. Also keep 独立 apart from 独自 and 单独, which mean physically alone rather than self-reliant - 他独自去了 is he went by himself, not he went independently.