I think if I was you, I would do something like this...
(this can be done outside python )
Select by location: polygons that contain points
--> create new layer Polys
##(Then inside python)
#OPEN PREVIOUSLY CREATED TEXT FILE
text_file = open("c:/AREAS.txt", "w")
#create python loop to grab the name, output the name to the text file
polys = c:/newpolygonlayer
Rows = gp.updatecursor(Polys)
Row = Rows.next()
for Row:
areaName = Row.TableField
print "We have selected " + str(areaName)
lines = [str(areaName) + "\n",]
text_file.writelines(lines)
text_file.close()
For the record I have never used the write to text function but I would like to one day. Also my syntax probably has a few errors but I hope this helps get the ball rolling.