Why aren't python nested functions called closures?

People are confusing about what closure is. Closure is not the inner function. the meaning of closure is act of closing. So inner function is closing over a nonlocal variable which is called free variable.

def counter_in(initial_value=0):
    # initial_value is the free variable
    def inc(increment=1):
        nonlocal initial_value
        initial_value += increment
        print(initial_value)
    return inc

when you call counter_in() this will return inc function which has a free variable initial_value. So we created a CLOSURE. people call inc as closure function and I think this is confusing people, people think "ok inner functions are closures". in reality inc is not a closure, since it is part of the closure, to make life easy, they call it closure function.

  myClosingOverFunc=counter_in(2)

this returns inc function which is closing over the free variable initial_value. when you invoke myClosingOverFunc

 myClosingOverFunc() 

it will print 2.

when python sees that a closure sytem exists, it creates a new obj called CELL. this will store only the name of the free variable which is initial_value in this case. This Cell obj will point to another object which stores the value of the initial_value.

in our example, initial_value in outer function and inner function will point to this cell object, and this cell object will be point to the value of the initial_value.

  variable initial_value =====>> CELL ==========>> value of initial_value

So when you call counter_in its scope is gone, but it does not matter. because variable initial_value is directly referencing the CELL Obj. and it indirectly references the value of initial_value. That is why even though scope of outer function is gone, inner function will still have access to the free variable

let's say I want to write a function, which takes in a function as an arg and returns how many times this function is called.

def counter(fn):
    # since cnt is a free var, python will create a cell and this cell will point to the value of cnt
    # every time cnt changes, cell will be pointing to the new value
    cnt = 0

    def inner(*args, **kwargs):
        # we cannot modidy cnt with out nonlocal
        nonlocal cnt
        cnt += 1
        print(f'{fn.__name__} has been called {cnt} times')
        # we are calling fn indirectly via the closue inner
        return fn(*args, **kwargs)
    return inner
      

in this example cnt is our free variable and inner + cnt create CLOSURE. when python sees this it will create a CELL Obj and cnt will always directly reference this cell obj and CELL will reference the another obj in the memory which stores the value of cnt. initially cnt=0.

 cnt   ======>>>>  CELL  =============>  0

when you invoke the inner function wih passing a parameter counter(myFunc)() this will increase the cnt by 1. so our referencing schema will change as follow:

 cnt   ======>>>>  CELL  =============>  1  #first counter(myFunc)()
 cnt   ======>>>>  CELL  =============>  2  #second counter(myFunc)()
 cnt   ======>>>>  CELL  =============>  3  #third counter(myFunc)()

this is only one instance of closure. You can create multiple instances of closure with passing another function

counter(differentFunc)()

this will create a different CELL obj from the above. We just have created another closure instance.

 cnt  ======>>  difCELL  ========>  1  #first counter(differentFunc)()
 cnt  ======>>  difCELL  ========>  2  #secon counter(differentFunc)()
 cnt  ======>>  difCELL  ========>  3  #third counter(differentFunc)()


  

For the readers of Structure and Interpretation of Computer Programs (SICP): there are 2 unrelated meanings of closure (CS VS Math), see Wikipedia for the latter/less common one:

Sussman and Abelson also use the term closure in the 1980s with a second, unrelated meaning: the property of an operator that adds data to a data structure to also be able to add nested data structures. This usage of the term comes from the mathematics usage rather than the prior usage in computer science. The authors consider this overlap in terminology to be "unfortunate."

• https://en.wikipedia.org/wiki/Closure_(computer_programming)#History_and_etymology

The second (mathematical) meaning is also used in SICP in Python, see for example the discussion of tuples

Our ability to use tuples as the elements of other tuples provides a new means of combination in our programming language. We call the ability for tuples to nest in this way a closure property of the tuple data type. In general, a method for combining data values satisfies the closure property if the result of combination can itself be combined using the same method.

• 2.3 Sequences | SICP in Python

Here is presented a way to identify if a function is a closure or not via code objects.

As already mentioned in other answers, not every nested function is a closure. Given a composite function (which represent the overall action) its intermediate states can be either be a closure or a nested function. A closure is a kind function which is "parametrized" by its (non-empty) enclosing scope, the space of free-variables. Notice that a composite function may be made by both types.

The (Python's) internal type code object represents the compiled function body. Its attribute co_cellvars and co_freevars can be used to "lookaround" the closure/scope of a function. As mentioned in the doc

• co_freevars: tuple of names of free variables (referenced via a function’s closure)
• co_cellvars: tuple of names of cell variables (referenced by containing scopes).

Once the function is read, by performing recursive calls a partial function is returned with its own __closure__ (hence cell_contents) and a list of free-variables from its clousre and in its scope.

Let introduce some support functions

# the "lookarounds"
def free_vars_from_closure_of(f):
    print(f.__name__, 'free vars from its closure',  f.__code__.co_cellvars)

def free_vars_in_scopes_of(f):
    print(f.__name__, 'free vars in its scope    ', f.__code__.co_freevars)

# read cells values
def cell_content(f):
    if f.__closure__ is not None:
        if len(f.__closure__) == 1: # otherwise problem with join
            c = f.__closure__[0].cell_contents
        else:
            c = ','.join(str(c.cell_contents) for c in f.__closure__)
    else:
        c = None

    print(f'cells of {f.__name__}: {c}')

Here an example from another answer rewritten in a more systematic way

def f1(x1):
    def f2(x2):
        a = 'free' # <- better choose different identifier to avoid confusion
        def f3(x3):
            return '%s %s %s %s' %  (x1, x2, a, x3)
        return f3
    return f2

# partial functions
p1 = f1('I')
p2 = p1('am')

# lookaround
for p in (f1, p1, p2):
    free_vars_in_scopes_of(p)
    free_vars_from_closure_of(p)
    cell_content(p)

Output

f1 free vars in its scope     ()         # <- because it's the most outer function
f1 free vars from its closure ('x1',)
cells of f1: None
f2 free vars in its scope     ('x1',)
f2 free vars from its closure ('a', 'x2')
cells of f2: I
f3 free vars in its scope     ('a', 'x1', 'x2')
f3 free vars from its closure ()        # <- because it's the most inner function
cells of f3: free, I, am

The lambda counterpart:

def g1(x1):
    return lambda x2, a='free': lambda x3: '%s %s %s %s' %  (x1, x2, a, x3)

From the point of view of the free variables/scoping are equivalent. The only minor differences are some values of some attributes of the code object: co_varnames, co_consts, co_code, co_lnotab, co_stacksize... and natuarlly the __name__ attribute.

A mixed example, closures and not at once:

# example: counter
def h1():             # <- not a closure
    c = 0
    def h2(c=c):      # <- not a closure
        def h3(x):    # <- closure
            def h4(): # <- closure
                nonlocal c
                c += 1
                print(c)
            return h4
        return h3
    return h2

# partial functions
p1 = h1()
p2 = p1()
p3 = p2('X')

p1() # do nothing
p2('X') # do nothing
p2('X') # do nothing
p3() # +=1
p3() # +=1
p3() # +=1

# lookaround
for p in (h1, p1, p2, p3):
    free_vars_in_scopes_of(p)
    #free_vars_from_closure_of(p)
    cell_content(p)

Output

1 X
2 X
3 X
h1 free vars in its scope     ()
cells of h1: None
h2 free vars in its scope     ()
cells of h2: None
h3 free vars in its scope     ('c',)
cells of h3: 3
h4 free vars in its scope     ('c', 'x')
cells of h4: 3,X

h1 and h2 are both not closures since they have no cell and no free-variables in their scope. h3 and h3 are closures and share (in this case) the same cell and free-variable for c. h4 has a further free-variable x with its own cell.

Final considerations:

• the __closure__ attribute and __code__.co_freevars can be used to check for values and names (identifiers) of the free-variables
• anti-analogies (in a broad sense) between nonlocal and __code__.co_cellvars: nonlocal acts towards the outer function, __code__.co_cellvars instead towards the internal function