I dont understand why the show was called the cosby show and not the huxtable show ???

https://www.youtube.com/watch?v=n3eJkf4erbI
https://www.youtube.com/watch?v=n3eJkf4erbI
https://www.youtube.com/watch?v=n3eJkf4erbI
https://www.youtube.com/watch?v=n3eJkf4erbI