How is set() implemented?

According to this thread:

Indeed, CPython's sets are implemented as something like dictionaries with dummy values (the keys being the members of the set), with some optimization(s) that exploit this lack of values

So basically a set uses a hashtable as its underlying data structure. This explains the O(1) membership checking, since looking up an item in a hashtable is an O(1) operation, on average.

If you are so inclined you can even browse the CPython source code for set which, according to Achim Domma, is mostly a cut-and-paste from the dict implementation.

When people say sets have O(1) membership-checking, they are talking about the average case. In the worst case (when all hashed values collide) membership-checking is O(n). See the Python wiki on time complexity.

The Wikipedia article says the best case time complexity for a hash table that does not resize is O(1 + k/n). This result does not directly apply to Python sets since Python sets use a hash table that resizes.

A little further on the Wikipedia article says that for the average case, and assuming a simple uniform hashing function, the time complexity is O(1/(1-k/n)), where k/n can be bounded by a constant c<1.

Big-O refers only to asymptotic behavior as n → ∞.Since k/n can be bounded by a constant, c<1, independent of n,

O(1/(1-k/n)) is no bigger than O(1/(1-c)) which is equivalent to O(constant) = O(1).

So assuming uniform simple hashing, on average, membership-checking for Python sets is O(1).

I think its a common mistake, set lookup (or hashtable for that matter) are not O(1).
from the Wikipedia

In the simplest model, the hash function is completely unspecified and the table does not resize. For the best possible choice of hash function, a table of size n with open addressing has no collisions and holds up to n elements, with a single comparison for successful lookup, and a table of size n with chaining and k keys has the minimum max(0, k-n) collisions and O(1 + k/n) comparisons for lookup. For the worst choice of hash function, every insertion causes a collision, and hash tables degenerate to linear search, with Ω(k) amortized comparisons per insertion and up to k comparisons for a successful lookup.

Related: Is a Java hashmap really O(1)?