Which of the following statements correctly opens a file in append mode without the need for specifying a close() function for the file?

The statement that opens a file in append mode while ensuring proper handling of file resources is written with the with statement in conjunction with the open function. Using with open('my_list.txt', 'a+') as file: effectively creates a context manager, which automatically manages the opening and closing of the file.

When using the with statement, Python takes care of closing the file once the block of code under the with is executed, regardless of whether an exception was raised or not. This is important as it helps prevent potential memory leaks or file corruption by ensuring that files are not left open unintentionally.

In this case, the 'a+' mode not only allows for appending new data to the end of the file but also enables reading from it, which adds flexibility in how the file can be used. The 'a' signifies append mode, and the '+' allows for reading.

The other options fail to correctly implement file opening in append mode or do not properly utilize the context manager pattern that ensures the file is automatically closed. This is why the choice that includes with and specifies as file is the correct and preferred approach.