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The complex coordinate isometry transports topology and affine segments

Statement

Equip $C$ with $d_{C} (z, w) = ∣ z - w ∣$ and $R^{2}$ with the Euclidean metric $d_{2}$ . The coordinate bijection $Φ (a + b i) = (a, b)$ is a bijective isometry and a homeomorphism. For every $z, w \in C$ and every $t \in R$ ,

$Φ ((1 - t) z + t w) = (1 - t) Φ (z) + t Φ (w) .$

For each $A \subseteq C$ , the image $Φ [A]$ is open, closed, bounded, compact, or connected exactly when $A$ has the corresponding property in the indicated ambient metric or topological space. Moreover,

$Φ [{int}_{d_{C}} A] = {int}_{d_{2}} Φ [A] .$

$Φ [{\overline{A}}^{d_{C}}] = {\overline{Φ [A]}}^{d_{2}} .$

$Φ [\partial_{d_{C}} A] = \partial_{d_{2}} Φ [A] .$

Subsets use their induced metric and topology.

Facts & Assumptions

Given: The coordinate map $Φ$ and the two ambient metric spaces.

[F1]

$C$ is the real coordinate plane, with coordinate arithmetic and The complex coordinate map preserves real scalar multiplication: $Φ$ is a bijection preserving addition and real scalar multiplication.

[F2]

The Euclidean metric, convergence, Cauchy sequences, and continuity on the complex plane and The $p$ -norms $∥ x ∥_{p}$ for rational $p \geq 1$ , and $∥ x ∥_{\infty}$ : $d_{C} (z, w) = ∣ z - w ∣ = ∥ Φ (z) - Φ (w) ∥_{2}$ .

[F3]

Open ball, closed ball and sphere in a metric space and The metric topology: a set is open when every one of its points has a ball around it inside the set; closed means open complement define balls and open and closed sets; Arbitrary unions and finite intersections of open sets are open, open balls are open and closed balls are closed establishes the metric topology; Isometry, isometric embedding, and the subspace metric on a subset defines isometry; Homeomorphism, open map, closed map, embedding, and what it means for a property to be topological defines homeomorphism.

[F4]

Interior, closure, boundary, limit point, isolated point and dense subset of a metric space: interior is tested by contained balls, closure by balls meeting the set, and boundary is closure minus interior.

[F5]

Bounded subset, diameter, distance from a point to a set, and distance between two sets in a metric space: a subset is bounded when it is empty or lies in some open ball.

[F6]

Open cover, subcover, compact metric space, and compact subset of a metric space and The image of a compact metric space under a continuous map is compact, and so is the image of any compact subset: compact subsets use the induced metric and continuous images of compact subsets are compact.

[F7]

A continuous image of a connected space is connected, and connectedness is a topological property: a continuous image of a connected subset, with its induced topology, is connected.

Proof

1.1F1F2F3

For $z, w \in C$ , [F2] gives $d_{2} (Φ z, Φ w) = d_{C} (z, w)$ , so bijectivity [F1] makes $Φ$ an isometry. For every $r > 0$ , $u \in B_{C} (z, r)$ exactly when $Φ u \in B_{2} (Φ z, r)$ ; hence $Φ [B_{C} (z, r)] = B_{2} (Φ z, r)$ , and the inverse has the reverse correspondence. If $U \subseteq C$ is open, let $B_{U}$ be the family of all open balls contained in $U$ . Every point of $U$ lies in one such ball by the definition of openness, and every member of $B_{U}$ is contained in $U$ , so $U = ⋃ B_{U}$ . The ball correspondence maps this union to a union of open Euclidean balls; the arbitrary-union clause of [F3] makes $Φ [U]$ open. The same argument for the inverse shows that both maps are open. Equality of distances gives continuity of both maps with $δ = ε$ , so the bijection is a homeomorphism by [F3].

1.2F1algebra

Real-linearity [F1] gives $Φ ((1 - t) z + t w) = Φ ((1 - t) z) + Φ (t w) = (1 - t) Φ (z) + t Φ (w)$ for every real $t$ .

2.1F3F4step 1.1

By the ball correspondence in step 1.1, a ball around $z$ is contained in $A$ exactly when its image ball is contained in $Φ [A]$ ; likewise a ball around $z$ meets $A$ exactly when its image ball meets $Φ [A]$ . The definitions in [F4] therefore give the interior and closure identities. Bijectivity preserves set difference, so it also gives the boundary identity. Openness transfers by the same ball test, and closedness transfers because complements correspond under the bijection.

3.1F5F6F7step 1.1given∎

If $A = \emptyset$ , then $Φ [A] = \emptyset$ and boundedness, compactness, and connectedness agree for the identical empty subsets. Otherwise, ball correspondence transfers containment in a ball in both directions, proving boundedness. The restrictions $Φ ∣_{A}$ and $Φ^{- 1} ∣_{Φ [A]}$ are continuous. Applying the compact-image theorem [F6] to these restrictions proves compactness in both directions; applying the connected-image theorem [F7] proves connectedness in both directions.

Depends on

Used by

Dependency tree · two levels

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Sources

J. Lebl, Guide to Cultivating Complex Analysis, Section 1.1.1 (standard reference, not scraped)
E. Stein and R. Shakarchi, Complex Analysis, Princeton Lectures in Analysis II, Chapter 1, Section 1.1 (standard reference, not scraped)