# Continuous binary operations on $\beta\mathbb{N}$

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.

• Compact Hausdorff spaces possess exponentials (set of all continuous maps with compact-open topology). Moreover (here I am not 100% confident, that's why I am not making this an answer), the exponential $X^{\beta\mathbb N}$ is homeomorphic to the cartesian power $X^{\mathbb N}$ for any compact Hausdorff $X$. It follows that continuous maps ${\beta\mathbb N}\times{\beta\mathbb N}\to X$ are in one-to-one correspondence with arbitrary sequences of continuous maps $f_1,f_2,...:{\beta\mathbb N}\to X$ – მამუკა ჯიბლაძე Dec 15 '18 at 16:47
• In other words (if the above is correct), any choice of values $f(i,j)$ on principal ultrafilters, (and these values might be arbitrary) extends uniquely to such an $f$. – მამუკა ჯიბლაძე Dec 15 '18 at 16:51
• ...which is impossible since ${\beta\mathbb N}\times{\beta\mathbb N}$ is not homeomorphic to $\beta({\mathbb N}\times{\mathbb N})$ - so I do make a mistake somewhere, sorry... – მამუკა ჯიბლაძე Dec 15 '18 at 16:58
• The usual extension of addition is not jointly contínuous for this reason – Benjamin Steinberg Dec 15 '18 at 21:52
• @მამუკაჯიბლაძე Although, when $X$ is compact Hausdorff, $X^{\beta\mathbb{N}}$ and $X^{\mathbb{N}}$ are isomorphic as sets, the topologies are different. The compact-open topology on $[0,1]^{\beta\mathbb{N}}$ is uniform convergence on compact sets, which by compactness of $\beta{\mathbb{N}}$ is just uniform convergence, whereas the topology coming from $[0,1]^{\mathbb{N}}$ is pointwise convergence on $\mathbb{N}$. – Robert Furber Dec 16 '18 at 4:12

In this paper, Dimension phenomena associated with $$\beta\mathbb{N}$$-spaces, Ilijas Farah proved that continuous maps from $$\beta\mathbb{N}^2$$ (and other powers) to $$\beta\mathbb{N}$$ are quite simple: there is a finite disjoint cover such that the map depends on one coordinate on each member of the cover.
• Thank you, KP! However, that's bitter news to me, unfortunately. Do you know whether there are similar results for operations on squares (that is, saying that operations $(\beta\mathbb{N})^2\to(\beta\mathbb{N})^2$ are also somehow simple)? – Damian Sobota Dec 16 '18 at 23:42
• Also: Malykhin, in $\beta\mathbb{N}$ (mathscinet.ams.org/mathscinet-getitem?mr=552052), shows that if $\prod_{n\in\omega}X_n$ contains a copy of $\beta\mathbb{N}$ then one of the factors already contains a copy. – KP Hart Dec 17 '18 at 9:04