Module is imported, but is still rendering NameError - python-3.x

Question

I am new to python packages and I am in process of learning how organize code into packages.
I tried out a small program for testing purpose and I am curious about an error that I encountered.
I have put the following package-structure (I am currently working with regular packages only as I don't have a good understanding of namespace packages):
parent/
main.py
p1/
__init__.py
The code in p1/__init__.py is as follows:
import math
print('p1 imported.')
The code in main.py is as follows:
from p1 import math
def main():
print(p1)
if __name__=='__main__':
main()
The output that it gives me is as follows:
path-to-parent>python3 main.py
p1 imported
Traceback (most recent call last):
File "main.py", line 12, in <module>
main()
File "main.py", line 8, in main
print(p1)
NameError: name 'p1' is not defined
I understand that it has imported math from sys.modules or the subsequent full search that it must have performed after encountering the import math statement in p1/__init__.py. It also executed the subsequent statement print('p1 imported') indicating that the module p1 has been imported. Then why does the NameError pop up?
With my reading of this page, my guess is that although the module p1 has been imported, it has not undergone the binding process because of the way it has been imported: from p1 import math.
I would still need help to understand this.
NOTE: I am using Anaconda Python 3.7.3 on windows 10 (64bit).

Answer 1

from p1 import math
def main():
print(p1)
if __name__=='__main__':
main()
p1 here is the name of the directory, it is not a python variable that can be used.
It is very roughly analogous to the following attempt:
with open('filename') as f:
print(filename)
Which will obviously raise NameError because filename is not defined.
Perhaps a better example is this:
from math import pi
print(math)
This also raises NameError because math is never defined as a variable that can be used outside of the import mechanism. math here is the name of the module from which pi is imported, but it can only be seen by the import mechanism.

Answer 2

It's much easier to demonstrate why what you're doing fails:
>>> from math import sin
>>> sin
<built-in function sin>
>>> math
Traceback (most recent call last):
File "<stdin>", line 1, in <module>
NameError: name 'math' is not defined
The point is, that math.sin is imported as sin, but the math module that is needed for this is not stored for reference anywhere. For that reason, there is no local math object you could refer to.
If you want to refer to the math module as a whole, just import it as a whole. For your example, just import p1.