definition of complex Euclidean topological space
Topics
About: topological space
The table of contents of this article
Starting Context
- The reader knows a definition of Euclidean topological space.
Target Context
- The reader will have a definition of complex Euclidean topological space.
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:
\( d\): \(\in \mathbb{N} \setminus \{0\}\)
\(*\mathbb{C}^d\): with the topology specified below
\( \mathbb{R}^{2 d}\): \(= \text{ the Euclidean topological space }\)
\( f\): \(: \mathbb{R}^{2 d} \to \mathbb{C}^d, (x^1, ..., x^{2 d}) \mapsto (x^1 + x^2 i, ..., x^{2 (d - 1) + 1} + x^{2 d} i)\)
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Conditions:
\(f \in \{\text{ the homeomorphisms }\}\)
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That means that the topology of \(\mathbb{C}^d\) is specified to make \(f\) homeomorphic.
2: Note
To be more specific, any \(S \subseteq \mathbb{C}^d\) is open if and only if \(f^{-1} (S) \subseteq \mathbb{R}^{2 d}\) is open.
As \(f\) is a bijection, it is indeed a topology.
