Question

I’m studying functional programming and lambda calculus but I’m wondering
if the closure term is also present in the Church’s original work or it’s a more modern
term strictly concerned to programming languages.

I remember that in the Church’s work there were the terms: free variable, closed into…,
and so on.

Answer 1

Consider the following function definition in Scheme:

(define (adder a)
  (lambda (x) (+ a x)))

The notion of explicit closure is not required in the pure lambda calculus, because variable substitution takes care of it. The above code snippet can be translated

λa λx . (a + x)

When you apply this to a value z, it becomes

λx . (z + x)

by β-reduction, which involves substitution. You can call this closure over a if you want.

(The example uses a function argument, but this holds true for any variable binding, since in the pure lambda calculus all variable bindings must occur via λ terms.)