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How do implicit relative imports work in Python?

Assume I have the following files,
pkg/
pkg/__init__.py
pkg/main.py # import string
pkg/string.py # print("Package's string module imported")
Now, if I run main.py, it says "Package's string module imported".
This makes sense and it works as per this statement in this link:
"it will first look in the package's directory"
Assume I modified the file structure slightly (added a core directory):
pkg/
pkg/__init__.py
plg/core/__init__.py
pkg/core/main.py # import string
pkg/string.py # print("Package's string module imported")
Now, if I run python core/main.py, it loads the built-in string module.
In the second case too, if it has to comply with the statement "it will first look in the package's directory" shouldn't it load the local string.py because pkg is the "package directory"?
My sense of the term "package directory" is specifically the root folder of a collection of folders with __init__.py. So in this case, pkg is the "package directory". It is applicable to main.py and also files in sub- directories like core/main.py because it is part of this "package".
Is this technically correct?
PS: What follows after # in the code snippet is the actual content of the file (with no leading spaces).

Packages are directories with a __init__.py file, yes, and are loaded as a module when found on the module search path. So pkg is only a package that you can import and treat as a package if the parent directory is on the module search path.
But by running the pkg/core/main.py file as a script, Python added the pkg/core directory to the module search path, not the parent directory of pkg. You do have a __init__.py file on your module search path now, but that's not what defines a package. You merely have a __main__ module, there is no package relationship to anything else, and you can't rely on implicit relative imports.
You have three options:
Do not run files inside packages as scripts. Put a script file outside of your package, and have that import your package as needed. You could put it next to the pkg directory, or make sure the pkg directory is first installed into a directory already on the module search path, or by having your script calculate the right path to add to sys.path.
Use the -m command line switch to run a module as if it is a script. If you use python -m pkg.core Python will look for a __main__.py file and run that as a script. The -m switch will add the current working directory to your module search path, so you can use that command when you are in the right working directory and everything will work. Or have your package installed in a directory already on the module search path.
Have your script add the right directory to the module search path (based on os.path.absolute(__file__) to get a path to the current file). Take into account that your script is always named __main__, and importing pkg.core.main would add a second, independent module object; you'd have two separate namespaces.
I also strongly advice against using implicit relative imports. You can easily mask top-level modules and packages by adding a nested package or module with the same name. pkg/time.py would be found before the standard-library time module if you tried to use import time inside the pkg package. Instead, use the Python 3 model of explicit relative module references; add from __future__ import absolute_import to all your files, and then use from . import <name> to be explicit as to where your module is being imported from.

Python3 Imports Issue

So... I am attempting to teach myself Python.
In such, I am attempting to build something that I appear to have no clue about...
I have a "workingdir" structure such as:
/
-- classes/
-- -- install
-- myfile
In myfile I am simply attempting to "import" the file install by using:
import classes.install
Which fails with: ImportError: No module named 'classes.install'
I have attempted the following as well, and all end the same way, with the same error:
import .classes.install
sys.path.append(os.path.join(os.path.dirname(__file__), '..'))
import classes.install
As well as putting an empty __init__.py file inside the classes directory
the file install simply contains:
class gyo_install():
inst = False
# check if we have everything we need installed.
def __init__():
print("Hello World")
What am I doing wrong? I've searched and searched and searched, everything I see points to the same solutions I've attempted, and none of them work.

Python looks for files with a .py extension when importing modules. So a file named myfile will not be recognized simply by the command import myfile. The pythonic way to ensure that the interpreter will find the module is to ensure it has a .py extension. Renaming myfile to myfile.py and install to install.py and then changing the import command to
import classes.install
should solve the problem.

Create __init__.py inside install directory.
Explanation: You can import from a file that is in your current directory or from a package. A package is a directory with __init__.py inside. In fact, a package can contain only this single file.
You can read the documentation for further information.

Import Error: No Module named common

My folder structure in pycharm is as follows.
--python
--concepts
--common
--myds.py
--__init__.py
--data_structures
--test_ds.py
I have the following line in test_ds.py
from common import my_ds
I get the following error.
ImportError: No module named 'common'
I have added common to Settings --> Project Interpreter -> Interpreter Paths
and the folder shows up as library root.
Still why am I getting this error.

Try from ..common import my_ds. Also make sure that it has an __init__.py file in that directory (not required but it's good practice).
As for the .. they indicate that you're importing from the parent package to the one you're currently on.

You need to make your common folder into a python package in order to import it in python. I think you've tried to do it and created init file in your common folder but actually it must be __init__.py. Rename it like this and then your package will be visible to python.
Hope it helps!

__init__ file doesn't work as expected in python

I have some folders and .py files in the following structure:
parent/
__init__.py
test.ipynb
code/
__init__.py
common.py
subcode/
__init__.py
get_data.py
In the __init__ file under the parent folder, I have import code and in the one of code, I have import subcode. But when I tried import code.subcode, I got such an error:
ImportError: No module named 'code.subcode'; 'code' is not a package
But when I just import code, no error is thrown. However, when I call code.subcode, this error happens:
AttributeError: module 'code' has no attribute 'subcode'
I try all of those mentioned above in the test.ipynb, which is at the root of the directory.
Do you know what is the reason and how can I fix it? Thanks!

The problem is that you are importing another module named code that is installed on your system rather than your own module. You can verify this by checking the module file path in code.__file__ after you import code.
The first thing to do is change the name of your module to avoid namespace collisions with the other code package on your system. If your new package name doesn't collide with something else, you should now either successfully be importing it and have it behave as expected, or it fails to import entirely.
If it fails to import, it is most likely because your parent directory is not in your PYTHONPATH environment variable.
There can potentially also be other more technical reasons that a module is not recognized by the interpreter such as old definitions being cached (in which case restarting the interpreter is often enough. Possibly after deleting any precompiled versions of the module). Another problem I have seen ended up being that a module contained a bug that made the interpreter unable to parse it. I am sure there are other odd possibilities out there.

You're on Python 3. You need to perform relative imports explicitly:
from . import code
The code module you're currently getting is the standard library code module.

Create and import a custom python package - import doesn't work in the root

I'm totally new to Python, and I want to create my first Python library for peronal uses.
I'm using Python 2.7.5, and running the IDLE interface.
So far, what I understood from the documentation and related questions is that:
Python goes through a list of directories listed in sys.path to find scripts and libraries
The package directory must contain a __init__.py file, which can be empty
The module I want to create should be a modulename.py file with the code inside the package directory
(Sources: http://www.johnny-lin.com/cdat_tips/tips_pylang/path.html --- https://docs.python.org/2/tutorial/modules.html)
And here is what I tried that fails:
Created a personal package directory C:\....\pythonlibs
Created a subpackage dir C:\....\pythonlibs\package
Created the __init__.py file inside both folders
Created the mymodule.py file in the packacge dir
And then in the IDLE used this code:
import sys
sys.path.append(r'C:\....\pythonlibs')
First issue:
Currently I have to do this append every time I enter the IDLE. How can I keep the directory in sys.path permanently just as there are a lot of other directories there?
Then I tried importing my package:
import pythonlibs #fails!! why?
import pythonlibs.package #fails!! why?
import package #works
The error is: ImportError: No module named pythonlibs
Second issue?
This seems to be against the documentation, why can't I import from the root pythonlibs folder?

With line
sys.path.append(r'C:\....\pythonlibs')
you are instructing interpreter to start looking for modules (libraries) in this directory. Since this directory does not contain pythonlibs folder (the parent does), it can't import it.
Similarly - because it contains the module package, it can import it.