from the Python documentation, I see that you can set methods for handling properties transparently:
class C(object):
    def __init__(self):
        self._x = None
    def getx(self):
        return self._x
    def setx(self, value):
        self._x = value
    def delx(self):
        del self._x
    x = property(getx, setx, delx, "I'm the 'x' property.")
Now assume that C._x is a list. In init it is simply set to []. So if I do the following:
c = C()
c.x = [1,2,3]
c.x will be set to [1,2,3]. What I want to be able to do now is
#...
c.x += 4
So that c.x is now [1,2,3,4]. The example is trivial, but obviously, I'd want the setx and getx methods to include some sort of processing and checks. Otherwise, using this approach would be silly. 
Edit: It might be good enough to just use the __add__ method on C to enforce the behaviour, but I wondered if it was possibly to put the behaviour on the property instead of the whole class
You can't overload operator for a specific attribute, because :
c.x += 4
# is equivalent to
c.x.__iadd__(4)
So in fact you are calling the __iadd__ operator of list. If you want to be able to do that, you have to create a new class, extending list, and overload operator __iadd__ or __add__.
class SuperList(list):
    def __iadd__(self, other):
        if type(other) == list or type(other) == SuperList:
            return super(SuperList, self).__iadd__(other)
        return super(SuperList, self).__iadd__([other])
                        No, that won't work.
Basically, here's what happens:
c.x += 4
This code is equivalent to this:
var temp = c.x
temp += 4
c.x = temp
However, temp here will be the list of 3 values, and this code doesn't work:
temp = [1, 2, 3]
temp += 4
You get:
TypeError: 'int' object is not iterable
So this has nothing to do with the property, and everything to do with the fact that the code needs to look like this:
temp += [4]
So this works:
c.x += [4]
This cannot:
c.x += 4
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