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?


About dominiczypen

I'm interested in general topology, order theory, and graph theory. This link takes you to my preprints on arXiv.
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