Completions of a metric space without choice

Masahiko Murakami

                        muramasa@ms.u-tokyo.ac.jp

Department of Mathematics, Hosei University

The definition of a completion of a metric space (Definition 2) seems queer, which is adopted in many books. The notion of dense is defined not merely by metric but by topology. Pure completion (Definition 2) seems more natural. We show that completion is not always unique upto isomorphism without the axiom of choice. In familiar method to have a completion, quotient of the space of the Cauchy sequences, we don’t have a complete metric space without choice (Example 2). Now, we don’t suppose the axiom of choice.

Definition 1. Let ⟨X, d⟩ be a metric space. a real valued function f on X is a location (see [1]) in X if the following conditions satisfy.

1) f(x) ≥ |f(y) − d(x, y)| for all x, y ∈ X.

2) inf_{x∈X} f(x) = 0.

We denote the set of the locations in X by X̂. For a ∈ X, we define the non-negative real valued function by d_{a}(x) = d(x, a). Define d̂: X̂ × X̂ → ℝ by d̂(f, g) = inf_{x∈X}(f(x) + g(x)).

Lemma 1. [1]

1) Any location f in X has at most one zero. If a ∈ X is a zero of f then f = d_{a}.

2) d̂(f, g) = sup_{x∈X} |f(x) − g(x)|.

3) d̂ is a metric on X̂.

4) d̂(f, d_{a}) = f(a).

5) The correspondence x ↦ d_{x} is an isometric embedding from X into X̂. That is d(x, y) = d̂(d_{x}, d_{y}) for x, y ∈ X.

We can identify a ∈ X and d_{a} ∈ X̂ by the lemma above. From now on, we admit X ⊆ X̂ in this way.

Theorem 2. The metric space (X̂, d̂) is complete.

Definition 2. A metric space Y is a completion of a metric space X, if Y is a complete metric space and X is a dense subspace of Y.

A metric space X is strongly complete if X itself is the only completion of X.

A metric space Y is a complete closure of a metric space X, if Y is a maximal completion of X, or Y is the only completion of X containing Y.

A metric space Y is a pure completion of a metric space X, if Y is a minimal complete metric superspace of X.

Lemma 3. Let X be a dense subspace of a metric space ⟨Y, d⟩. Let φ be a map from Y defined by φ(y) = d_{y}↾X. Then φ is an isometric embedding into X̂ and φ↾X is the identity.

Theorem 4 (Existence of complete closure). X̂ is a complete closure of X.

Theorem 5 (Uniqueness of complete closure). There is a unique complete closure of an metric space ⟨X, d⟩ upto isomorphism.

Theorem 6 (Unique existence of pure completion). We denote

X^{♮} = ⋂ {Y | X ⊆ Y ⊆ X̂, Y is complete}

Then X^{♮} is a pure completion of X. Pure completion is unique upto isomorphism.

For a subset A ⊆ X̂, We denote

Ã = {a ∈ X̂ | there is a Cauchy sequence in A which converges to a}

We see that X ⊆ X̃ ⊆ X^{♮} ⊆ X̂. In ZF (without choice), we cannot prove any equality of the relation above by the following examples. Example 1. Let D_{1} be a Dedekind-finite infinite subset of interval (0, 1] with inf D_{1} = 0. Then 0 ∉ D_{1} = D̃_{1} and 0 ∈ D̂_{1}. In this case D_{1} = D̃_{1} = D_{1}^{♮} ≠ D̂_{1}. Example 2. For x ∈ D_{1}, define S_{x} ⊆ ℝ^{2} by

S_{x} = \{(xy + \frac{1 − y}{n + 1}, y) | 0 < y ≤ 1\} if n ∈ N, \frac{1}{n + 2} < x ≤ \frac{1}{n + 1}.

Put D_{2} = ⋃ {S_{x} | x ∈ D_{1}}. Then (0, 0) ∉ D̃_{2}, (0, 0) ∈ ˜ D̃_{2} ⊆ D_{2}^{♮}, (0, 1) ∉ D_{2}^{♮} and (0, 1) ∈ D̂_{2}. In this case D_{2} ≠ D̃_{2} ≠ D_{2}^{♮} ≠ D̂_{2} and D̃_{2} is not complete.

Proposition 7. Define inductively the function F from the ordinals by F (ζ) = X ∪ ⋃_{ξ<ζ} F̃ (ξ). Then there is an ordinal κ such that X^{♮} = F (κ).

References

[1] F. Richman, The fundamental theorem of algebra: a constructive development without choice, Pacific Journal of math., 196 No.1 (2000), 213–230.

[2] Thomas J. Jech, The axiom of choice, North-Holland, Amsterdam, (1973).