description/proof of that for continuous map from topological space into \(1\)-dimensional complex Euclidean topological space, its absolute map is continuous
Topics
About: topological space
The table of contents of this article
Starting Context
- The reader knows a definition of continuous, topological spaces map.
- The reader knows a definition of complex Euclidean topological space.
- The reader admits the proposition that for any maps between any arbitrary subspaces of any topological spaces continuous at any corresponding points, the composition is continuous at the point.
Target Context
- The reader will have a description and a proof of the proposition that for any continuous map from any topological space into the \(1\)-dimensional complex Euclidean topological space, its absolute map is continuous.
Orientation
There is a list of definitions discussed so far in this site.
There is a list of propositions discussed so far in this site.
Main Body
1: Structured Description
Here is the rules of Structured Description.
Entities:
\(T\): \(\in \{\text{ the topological spaces }\}\)
\(\mathbb{C}\): \(= \text{ the complex Euclidean topological space }\)
\(f\): \(: T \to \mathbb{C}\), \(\in \{\text{ the continuous maps }\}\)
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Statements:
\(\vert f \vert: T \to \mathbb{R}, t \mapsto \vert f (t) \vert \in \{\text{ the continuous maps }\}\)
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2: Proof
Whole Strategy: Step 1: take \(g_1: \mathbb{C} \to \mathbb{R} \times \mathbb{R}, a + b i \mapsto (a, b)\), \(g_2: \mathbb{R} \times \mathbb{R} \to \mathbb{R}, (a, b) \mapsto a^2 + b^2\), and \(g_3: [0, \infty) \subseteq \mathbb{R} \to \mathbb{R}, r \mapsto \sqrt{r}\), and see that they are continuous; Step 2: see that \(\vert f \vert = g_3 \circ g_2 \circ g_1 \circ f\); Step 3: conclude the proposition.
Step 1:
Let us take \(g_1: \mathbb{C} \to \mathbb{R} \times \mathbb{R}, a + b i \mapsto (a, b)\).
Let us take \(g_2: \mathbb{R} \times \mathbb{R} \to \mathbb{R}, (a, b) \mapsto a^2 + b^2\).
Let us take \(g_3: [0, \infty) \subseteq \mathbb{R} \to \mathbb{R}, r \mapsto \sqrt{r}\).
\(g_1\) is continuous, by the definition of complex Euclidean topological space.
\(g_2\) is continuous, as a well-known fact.
\(g_3\) is continuous, as a well-known fact.
Step 2:
Let us see that \(\vert f \vert = g_3 \circ g_2 \circ g_1 \circ f\).
For each \(t \in T\), \(g_3 \circ g_2 \circ g_1 \circ f: t \mapsto f (t) = a + b i \mapsto (a, b) \mapsto a^2 + b^2 \mapsto \sqrt{a^2 + b^2} = \vert f (t) \vert\).
Step 3:
\(\vert f \vert\) is continuous, by the proposition that for any maps between any arbitrary subspaces of any topological spaces continuous at any corresponding points, the composition is continuous at the point.
