How Do You Count Unique Words in a Document with Python

I am Python newbie trying to understand the answer given here to the question of counting unique words in a document. The answer is:

print len(set(w.lower() for w in open('filename.dat').read().split()))

Reads the entire file into memory, splits it into words using whitespace, converts each word to lower case, creates a (unique) set from the lowercase words, counts them and prints the output

To try understand that, I am trying to implement it in Python step by step. I can import the text tile using open and read, divide it into individual words using split, and make them all lower case using lower. I can also create a set of the unique words in the list. However, I cannot figure out how to do the last part - count the number of unique words.

I thought I could finish by iterating through the items in the set of unique words and counting them in the original lower-case list, but I find that that the set construct is not indexable.

So I guess I am trying to do something that in natural language is like, for all the items in the set, tell me how many times they occur in the lower case list. But I cannot quite figure out how to do that, and I suspect some underlying misunderstanding of Python is holding me back.

#python