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Compact normalized compressed phase sets over bounded regions

Statement

Let $Ω$ be a smooth Euclidean exterior with the compressed-ray convention of Independent generalized reflected-ray and GCC convention. For a compact $K \subset \overline{Ω}$ , the compressed image $K$ of ${(0, x, - 1, ξ) : x \in K, ∣ ξ ∣ = 1}$ is compact and metrizable. Its base projection is continuous. Compression identifies the two normal characteristic directions at a hyperbolic boundary point; compactness does not select a preferred outgoing ray.

Facts & Assumptions

[F1]

The compressed-bundle topology and compression map are those of Independent generalized reflected-ray and GCC convention. On a spatial boundary chart $U$ , at fixed time $t = 0$ , the local compressed coordinates are $(y (x); η^{'}, y_{n} η_{n})$ , with $η^{'}$ including temporal frequency. On an interior chart use the ordinary cotangent coordinates. In either case write $c_{U} (q)$ for the fiber-coordinate vector, so $(x (q), c_{U} (q))$ is a coordinate description of the compressed space over $U$ . Its topology is the subspace topology in these coordinates; the base map $q \mapsto x (q)$ and $j$ are continuous there. The normalization $τ = - 1$ fixes the forward unit-speed convention.

[F2]

Heine–Borel, Heine-Borel in ℝⁿ: with the Euclidean metric a subset of ℝⁿ is compact if and only if it is closed and bounded, and the proof by bisection uses no choice principle; the same holds on the real line, identifies compact Euclidean sets with closed bounded sets. In particular the given compact base set has the finite-open-subcover property, and its product with the unit sphere is closed and bounded in Euclidean space.

[F3]

The continuous-image theorem The image of a compact metric space under a continuous map is compact, and so is the image of any compact subset applies to a continuous map between metric spaces with compact domain. We will establish a compatible metric on the compressed image before applying it.

Proof

Given: A compact $K \subset \overline{Ω}$ and the normalized phase set $S = {(0, x, - 1, ξ) : x \in K, ∣ ξ ∣ = 1}$ , with compressed image $K = j (S)$ . Goal: A metric inducing the compressed topology on $K$ , compactness in that topology, and continuity of base projection.

1.1givenF1F2

If $K$ is empty, its compressed image is empty and all conclusions hold. Otherwise, $S$ is a closed bounded subset of a finite-dimensional Euclidean space and is compact by [F2]. Put $X = \overline{Ω}$ . Consider all interior or boundary chart neighborhoods $U \subset X$ together with $a \in K$ and $r > 0$ such that $X \cap \overline{B} (a, 2 r) \subset U$ . The sets $X \cap B (a, r)$ in this family cover $K$ , because chart neighborhoods are relatively open in $X$ . Compactness of $K$ retains finitely many such triples $(U_{i}, a_{i}, r_{i})$ , $1 \leq i \leq m$ . This uses compactness of the base, not of $K$ ; taking all admissible triples and then a finite subcover requires no infinite choice.

2.1F1step 1.1construct

Define $χ_{i} (x) = \max {0, 1 - ∣ x - a_{i} ∣ / r_{i}}$ for $x \in X$ . These continuous functions have support in $X \cap \overline{B} (a_{i}, r_{i}) \subset U_{i}$ , and at every $x \in K$ some $χ_{i} (x)$ is positive. For $q \in K$ define $g_{i} (q) = χ_{i} (x (q)) c_{U_{i}} (q)$ when $x (q) \in U_{i}$ , and $g_{i} (q) = 0$ otherwise. On the part over $U_{i}$ this is continuous by [F1]. At a base point outside $U_{i}$ , the base lies outside the closed set $X \cap \overline{B} (a_{i}, r_{i})$ , so a base neighborhood misses the support and $g_{i}$ is identically zero above it. Hence each $g_{i}$ is continuous globally on $K$ . Continuity of the base map here follows locally from [F1], without assuming a global metric.

3.1F1step 2.1

Define $E (q) = (x (q), g_{1} (q), \dots, g_{m} (q))$ , taking values in a finite-dimensional Euclidean space. It is continuous by step 2.1. If $E (q) = E (\tilde{q})$ , then both points have the same base $x$ . Choose an index $i$ with $χ_{i} (x) > 0$ . Dividing the equal $i$ th blocks by this positive number gives $c_{U_{i}} (q) = c_{U_{i}} (\tilde{q})$ , so the local coordinates agree and $q = \tilde{q}$ . Thus $E$ is injective.

4.1F1step 2.1step 3.1

On the relatively open part of $E (K)$ where $χ_{i} (x) > 0$ , its inverse is continuous: recover the compressed chart coordinates as $(x, g_{i} / χ_{i} (x))$ and then use the inverse coordinate chart from [F1]. These parts cover $E (K)$ , so the inverse of $E$ is continuous everywhere. Consequently $d (q, \tilde{q}) = ∣ E (q) - E (\tilde{q}) ∣$ is a metric on $K$ inducing exactly its compressed topology. The metric axioms follow from the Euclidean norm and injectivity; equality of topologies follows from the two continuous inverse maps. This supplies global separation and topology compatibility explicitly.

5.1F1F3step 1.1step 3.1step 4.1∎

Compression $j : S \to K$ is continuous in the original compressed topology by its local coordinate formula in [F1]. Step 4.1 identifies this topology with a metric topology, and step 1.1 supplies a compact metric domain. The hypotheses of [F3] therefore hold, proving compactness of $K$ . Its base projection is continuous, also as the first coordinate of $E$ . At a hyperbolic boundary point, the two characteristic normal covectors have the same tangential components and opposite normal components; multiplication by $y_{n} = 0$ identifies them under $j$ . The construction concerns phase points only and makes no selection among outgoing rays.

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