Genericity and category This paper by Ambos-Spies and Mayordomo on the theory of algorithmic randomness introduces the notion of genericity saying that it is based on Baire category while the usual notion of randomness is build up on the measure theory.  However, it does not state how the topology of the Cantor space is actually connected with the definition of generic sequences; nor does the famous textbook on the subject by Rodney and Hirschfeldt.
I would be most grateful if you could provide information about the resources that discuss the connection between genericity and Baire category in detail.
 A: I am not sure if this answers your question:
Within a 'randomness viewpoint', the sets of measure 1 are considered 'big', whereas within a 'genericity viewpoint', the comeager sets are considered 'big'. In both cases we study sequences which are in 'sufficiently many' big sets.
'sufficiently many' has to be countable to ensure the existence of elements in 'sufficiently many' big sets: The intersection of countably many sets of measure 1 is still of measure 1, and the intersection of countably many comeager set is still comeager.
So for genericity in the Cantor space, we first identify a countable collection of dense open sets and we say that a sequence is generic if it belongs to all of them.
Here are a few examples of some genericity notions which have been studied :


*

*The sequences that are in all the dense open sets of a countable model of set theory are the Cohen-generic sequences.

*The sequences that are in all the computably enumerable dense open set of the Cantor space (with the usual topology) are the weakly-1-generic sequences.

*The sequences which are in all the sets $U \cup (U^c)^\circ$ for all computably enumerable open sets $U$ are the 1-generics sequences.
In the last example $(U^c)^\circ$ denotes the interior of the complement of $U$. Note that for any open set $U$ we have that $U \cup (U^c)^\circ$ is a dense open set.
