It is well-known that the operation of addition of two ultrafilters on the set $\mathbb{N}$ of natural numbers which extends the natural addition on $\mathbb{N}$ to $\beta\mathbb{N}$, the Cech-Stone compactification of $\mathbb{N}$, is not continuous (it is only right-continuous).

I am thus looking for examples of continuous binary operations on $\beta\mathbb{N}$ i.e. such functions $\beta\mathbb{N}\times\beta\mathbb{N}\to\beta\mathbb{N}$ which are (jointly) continuous (not only separately continuous). I am only interested in operations that engage both variables (so e.g. projections and their compositions with mappings $\beta\mathbb{N}\to\beta\mathbb{N}$ drop out), but they need not have any additional properties like associativity etc.

arbitrarysequences of continuous maps $f_1,f_2,...:{\beta\mathbb N}\to X$ $\endgroup$ – მამუკა ჯიბლაძე Dec 15 '18 at 16:47