Wulliam became famous and had a chance to study in high school again.
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In this sentence, what is the role of "to....." in "a chance to..."?
I mean the relationship between 'a chance' and "the to infinitive." Are they in apposition? Or does "the to infinitive" modify 'chance' and have the role of the adjective use?
I'd say the infinitive phrase is acting as an adjective here, telling what kind of chance it is.
In the sentence "Wulliam became famous and had a chance to study in high school again," the phrase "to study in high school again" functions as an infinitive phrase, specifically an infinitive of purpose.
The phrase "to study" serves as the infinitive verb, and "in high school again" acts as an adverbial phrase, providing additional information about the location or context of the action. Together, they modify the noun "chance" and represent the purpose or intention of the chance that Wulliam had.
So, to answer your question directly, the "to infinitive" phrase does not modify "chance" as an adjective would, but rather functions to explain the purpose or intention of the chance. The relationship between "a chance" and "to study in high school again" can be seen as a noun phrase modified by an infinitive of purpose. They are not in apposition because they do not serve the same grammatical function.