If IEEE floating-point numbers had countably infinite precisions, its domain would be: $$ \{-\infty\}\cup\mathbb{R}^-\cup\{-0,+0\}\cup\mathbb{R}^+\cup\{+\infty\}\cup\{\text{NaN}\} $$
Let's denote this $\mathbb{FP}$ and give $\mathbb{FP}^2$ the product topology.
The addition is defined as:
- $+0 + \pm0 = +0$
- $-0 + \pm0 = \pm0$ (double sign in same order)
- $x + (-x) = +0$ for $x \in \mathbb{R}^+$
- $\pm\infty + x = \pm\infty$ for $x \in \mathbb{R}$(double sign in same order)
- $\pm\infty + \pm\infty = \pm\infty$ (double sign in same order)
- $\pm\infty + \mp\infty = \text{NaN}$ (double sign in same order)
- If either operand is NaN, the result is NaN.
- Addition is commutative.
- Otherwise, the usual addition in $\mathbb{R}$
The negation is defined as
- $-(\pm0) = \mp0$ (double sign in same order)
- $-(\pm\infty) = \mp\infty$ (double sign in same order)
- $-\text{NaN} = \text{NaN}$
- Otherwise, the usual negation in $\mathbb{R}$
Assume both addition and negation are continuous, as long as the subspaces $\mathbb{R}^+$ and $\mathbb{R}^-$ maintain the usual topology.
The main question is: Are the zeros topologically distinguishable?
Note: I actually want the "prettiest" topology that explains the behavior of IEEE floating-point numbers. This is such a subjective notion, yet there are something I definitely don't want:
Trivial topology (a separation axiom would prevent it)
Discrete topology (connectedness or compactness would prevent it)
I wanted the $T_1$ axiom so badly, but I couldn't separate the zeros. That's where the above question emerged.
As a consequence, I want the topology to satisfy as many space properties as possible.
So there is an additional question. If the answer is "yes", what is the maximum number of satisfiable-at-once conditions below?:
Addition is quotient. (perfectness is preferable)
Negation is quotient. (perfectness is preferable)
$\mathbb{FP}$ is $T_1$. (stronger separation axioms are preferable)
$\mathbb{FP}$ is metrizable. (complete matrizability is preferable)
$\mathbb{FP}$ is connected. (path-connectedness is preferable)
$\mathbb{FP}$ is Lindelöf. (compactness is preferable)
$\mathbb{FP}$ is second-countable. (first-countablility is preferable)
$\mathbb{FP}$ has a countable dense subset.
I might've also concerned about other operations, such as multiplication, reciprocal, absolute value function, signum function, transcendental functions, or special functions. But that really should be a story for another day...