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The Hilbert curve is a continuous space-filling curve that maps $[0, 1]$ to $[0, 1]^n$. Like all other space-filling curves, it is not one-to-one. I am wondering if the Hilbert curve, or a common variant thereof, becomes a continuous bijective map if we restrict its domain/range to $\mathbb{CN} \mapsto \mathbb{CN}^n$ where $\mathbb{CN} \subset [0, 1]$ denotes the set of computable numbers that lie within the unit interval.

At first glance, this seems like a true statement because we have the following two algorithms [1]:

• A forward algorithm that takes any number $t \in \mathbb{CN}$ as input, iterates over its binary digits, and sequentially produces the binary digits of all numbers in a corresponding (unique) tuple $T \in \mathbb{CN}^n$.
• A reverse algorithm that takes any tuple $T \in \mathbb{CN}^n$ as input, (simultaneously) iterates over the binary digits of $T$'s elements, and sequentially produces the binary digits of a (unique) number $t \in \mathbb{CN}$.

However, I'm not sure if this simple reasoning neglects to account for any of the finer points in computable analysis, especially with respect to the curve's continuity.

[1] The algorithms I mention above is described here. As far as I can tell, these algorithms operate on the standard Hilbert curve and utilize Gray codes to construct outputs from given inputs.

Thanks in advance for any comments, ideas and pointers to the relevant literature.

The Hilbert curve is a continuous space-filling curve that maps $I$ to $I^n$ where $I$ denotes the unit interval [1]. Like all other space-filling curves, it is not one-to-one. I am wondering if the Hilbert curve, or a common variant thereof, becomes a continuous bijective map if we restrict its domain/range to $\mathbb{CN} \mapsto \mathbb{CN}^n$ where $\mathbb{CN} \subset I$ denotes the set of computable numbers that lie within the unit interval.

At first glance, this seems like a true statement because we have the following two algorithms [2]:

• A forward algorithm that takes any number $t \in \mathbb{CN}$ as input, iterates over its binary digits, and sequentially produces the binary digits of all numbers in a corresponding (unique) tuple $T \in \mathbb{CN}^n$.
• A reverse algorithm that takes any tuple $T \in \mathbb{CN}^n$ as input, (simultaneously) iterates over the binary digits of $T$'s elements, and sequentially produces the binary digits of a (unique) number $t \in \mathbb{CN}$.

However, I'm not sure if this simple reasoning neglects to account for any of the finer points in computable analysis, especially with respect to the curve's continuity.

[1] As Pietro Majer points out, the original Hilbert curve maps $I$ to $I^2$. In this question, I am referring to its standard n-dimensional extension that uses Gray codes to construct its finite approximations.

[2] The algorithms I mention above is described here. As far as I can tell, these algorithms operate on the standard n-dimensional Hilbert curve (and utilize Gray codes to construct outputs from given inputs).

Thanks in advance for any comments, ideas and pointers to the relevant literature.

A Hilbert curve is a continuous space-filling curve that maps $[0, 1]$ to $[0, 1]^n$. Like all other space-filling curves, it is not one-to-one. I am wondering if a Hilbert curve becomes a continous bijective map if we restrict its domain/range to $\mathbb{CN} \mapsto \mathbb{CN}^n$ where $\mathbb{CN} \subset [0, 1]$ denotes the set of computable numbers that lie within the unit interval.

At first glance, this seems like a true statement because we have the following two algorithms:

• A forward algorithm that takes any number $t \in \mathbb{CN}$ as input, iterates over its binary digits, and sequentially produces the binary digits of all numbers in a corresponding (unique) tuple $T \in \mathbb{CN}^n$.
• A reverse algorithm that takes any tuple $T \in \mathbb{CN}^n$ as input, (simultaneously) iterates over the binary digits of $T$'s elements, and sequentially produces the binary digits of a (unique) number $t \in \mathbb{CN}$.

However, I'm not sure if this simple reasoning neglects to account for any of the finer points in computable analysis, especially with respect to the curve's continuity.

Note: The specific curve in question, along with the algorithms I mention above, is described here.

Thanks in advance for any comments, ideas and pointers to the relevant literature.

The Hilbert curve is a continuous space-filling curve that maps $[0, 1]$ to $[0, 1]^n$. Like all other space-filling curves, it is not one-to-one. I am wondering if the Hilbert curve, or a common variant thereof, becomes a continuous bijective map if we restrict its domain/range to $\mathbb{CN} \mapsto \mathbb{CN}^n$ where $\mathbb{CN} \subset [0, 1]$ denotes the set of computable numbers that lie within the unit interval.

At first glance, this seems like a true statement because we have the following two algorithms [1]:

• A forward algorithm that takes any number $t \in \mathbb{CN}$ as input, iterates over its binary digits, and sequentially produces the binary digits of all numbers in a corresponding (unique) tuple $T \in \mathbb{CN}^n$.
• A reverse algorithm that takes any tuple $T \in \mathbb{CN}^n$ as input, (simultaneously) iterates over the binary digits of $T$'s elements, and sequentially produces the binary digits of a (unique) number $t \in \mathbb{CN}$.

However, I'm not sure if this simple reasoning neglects to account for any of the finer points in computable analysis, especially with respect to the curve's continuity.

[1] The algorithms I mention above is described here. As far as I can tell, these algorithms operate on the standard Hilbert curve and utilize Gray codes to construct outputs from given inputs.

Thanks in advance for any comments, ideas and pointers to the relevant literature.

A Hilbert curve is a continuous space-filling curve that maps $[0, 1]$ to $[0, 1]^n$. Like all other space-filling curves, it is not one-to-one. I am wondering if a Hilbert curve becomes a continous bijective map if we restrict its domain/range to $\mathbb{CN} \mapsto \mathbb{CN}^n$ where $\mathbb{CN} \subset [0, 1]$ denotes the set of computable numbers that lie within the unit interval.

At first glance, this seems like a true statement because we have an algorithm that takes any $t \in \mathbb{CN}$, iterates over its binary digits, and sequentially produces the binary digits of all numbers in its corresponding (unique) tuple $T \in \mathbb{CN}^n$. However, I'm not sure if this simple reasoning neglects to account for any of the finer points in computable analysis, especially with respect to the curve's continuity.

Note: The actual curve, along with the algorithm I mention above, is described here.

Thanks in advance for any comments, ideas and pointers to the relevant literature.

A Hilbert curve is a continuous space-filling curve that maps $[0, 1]$ to $[0, 1]^n$. Like all other space-filling curves, it is not one-to-one. I am wondering if a Hilbert curve becomes a continous bijective map if we restrict its domain/range to $\mathbb{CN} \mapsto \mathbb{CN}^n$ where $\mathbb{CN} \subset [0, 1]$ denotes the set of computable numbers that lie within the unit interval.

At first glance, this seems like a true statement because we have the following two algorithms:

• A forward algorithm that takes any number $t \in \mathbb{CN}$ as input, iterates over its binary digits, and sequentially produces the binary digits of all numbers in a corresponding (unique) tuple $T \in \mathbb{CN}^n$.
• A reverse algorithm that takes any tuple $T \in \mathbb{CN}^n$ as input, (simultaneously) iterates over the binary digits of $T$'s elements, and sequentially produces the binary digits of a (unique) number $t \in \mathbb{CN}$.

However, I'm not sure if this simple reasoning neglects to account for any of the finer points in computable analysis, especially with respect to the curve's continuity.

Note: The specific curve in question, along with the algorithms I mention above, is described here.

Thanks in advance for any comments, ideas and pointers to the relevant literature.