# computability – Is it valid to make an admission of a topological space by a “partial quotient map”?

It is well-known that the Sierpiński space, $${0,1}$$ endowed with topology $${emptyset, {0},{0,1}}$$, is admissible. I tried to implement it in Haskell.

First I implement $$mathbb{N}$$ (including zero; some topologists might prefer denoting it by $$S_omega$$) by Peano definition:

``````data Peano = Zero | Succ Peano deriving (Eq, Ord)
``````

This encodes $$mathbb{N}$$, but there is one additional value lurking behind: `fix Succ`. I denote it by $$omega$$, and I denote $$mathbb{N} cup {omega}$$ by $$overline{S_omega}$$. We observe that $$overline{S_omega}$$ is in order topology.

I abuse this fact and implement the Sierpiński space by taking a quotient space. The quotient map $$q$$ is:

$$q(o) = begin{cases} 0 & text{if } o < omega \ 1 & text{o.w.} end{cases}$$

In Haskell, This can be realized by:

``````newtype Sierpinski = Sierpinski Peano

instance Eq Sierpinski where
Sierpinski m == Sierpinski n = let
q m = case m of
Zero   -> False
Succ n -> q n
in q m == q n
``````

But to think about it, `q` doesn’t halt on `fix Succ`. In other words, `q` is partial, and doesn’t match $$q$$. Is this really a valid implementation of the Sierpiński space?