Let $S(\mathbb{N}\times\mathbb{N}, \mathbb{N})$ the set of bijective functions from $\mathbb{N}\times\mathbb{N}$ to $\mathbb{N}$. I have some questions about this set.

Note that $S(\mathbb{N}\times\mathbb{N}, \mathbb{N}) \subseteq \mathbb{N}^{\mathbb{N}\times\mathbb{N}}$. Consider the discrete topology in $\mathbb{N}$, and $\mathbb{N}^{\mathbb{N}\times\mathbb{N}}$ with product topology.

  • what are the autohomomorphisms of $S(\mathbb{N}\times\mathbb{N}, \mathbb{N})$.
  • It is possible to define an operation on $S(\mathbb{N}\times\mathbb{N}, \mathbb{N})$, such that it is a semigroup (or group).

The reason for my questions is that we know several things about $S_\infty(\mathbb{N})$ (set of bijective functions from $\mathbb{N}$ to $\mathbb{N}$), I was wondering if it is possible to transfer some results from $S_\infty(\mathbb{N})$, to $S(\mathbb{N}\times\mathbb{N}, \mathbb{N})$. At least topologically $S_\infty(\mathbb{N})$ and $S(\mathbb{N}\times\mathbb{N}, \mathbb{N})$ are the same, but I can't see what the autohomeomorphisms of $S(\mathbb{N}\times\mathbb{N}, \mathbb{N})$ look like, for example.

  • 3
    $\begingroup$ There's no reason to consider $\mathbb{N}\times\mathbb{N}$, since $S(\mathbb{N}\times \mathbb{N}, \mathbb{N})\cong S(\mathbb{N}, \mathbb{N}) = S_\infty(\mathbb{N})$ using the bijection of $\mathbb{N}\times\mathbb{N}$ with $\mathbb{N}$. This also means that the set of autohomeomorphisms will be the same, up to application of this homeomorphism. $\endgroup$
    – Jakobian
    Nov 20 at 20:57


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