## Two notions of continuity in the space of natural sequences

In this presentation by Martín Escardó I came across a nice concept of continuity in the set of all functions $(\mathbb{N} \to \mathbb{N}) := \mathbb{N}^\mathbb{N}$.

For $n\in \mathbb{N}$ and $\alpha, \beta \in (\mathbb{N}\to \mathbb{N})$ we say that

$\alpha =_{(n)} \beta$ if and only if for all $i\leq n$ we have $\alpha(i) = \beta(i)$.

Then we define $f: (\mathbb{N}\to \mathbb{N}) \to (\mathbb{N}\to \mathbb{N})$ to be finitely continuous if

$\forall \alpha \in (\mathbb{N}\to \mathbb{N}).\forall n\in \mathbb{N}. \exists m\in \mathbb{N}. \forall \beta \in (\mathbb{N}\to \mathbb{N}).$

$(\alpha =_{(m)}\beta\implies f(\alpha) =_{(n)} f(\beta)).$

So if you want to know the first $n$ positions of the output sequence $f(\alpha)$, it suffices to know the first $m$ positions of the input sequence $\alpha$.

It turns out that this definition is equivalent to topological equivalence of functions $f: (\mathbb{N}\to \mathbb{N}) \to (\mathbb{N}\to \mathbb{N})$, where $\mathbb{N}$ is endowed with the discrete topology and $(\mathbb{N}\to \mathbb{N})$ carries the interval topology.

Nice, isn’t it?