definition of quaternion Euclidean topological space
Topics
About: topological space
The table of contents of this article
Starting Context
- The reader knows a definition of quaternions division associative algebra.
- The reader knows a definition of Euclidean topological space.
- The reader admits the proposition that the quaternions division associative algebra is 'rings - homomorphisms' isomorphic to a subring of the \(2 \times 2\) complex matrices ring.
Target Context
- The reader will have a definition of quaternion 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:
\( d\): \(\in \mathbb{N} \setminus \{0\}\)
\(*\mathbb{H}^d\): with the topology specified below
\( M_{2 \times 2 d} (\mathbb{C})\): \(= \text{ the complex matrices topological space }\)
\( f\): \(: \mathbb{H} \to R\), \(= \text{ the 'rings - homomorphisms' isomorphism }\) defined in the proposition that the quaternions division associative algebra is 'rings - homomorphisms' isomorphic to a subring of the \(2 \times 2\) complex matrices ring
\( f'''\): \(: \mathbb{H}^d \to M_{2 \times 2 d} (\mathbb{C})\) which replaces each component of each \(p \in \mathbb{H}^n\) regarded as the row with \(f (p_j)\)
\( f'''^`\): \(: \mathbb{H}^d \to f''' (\mathbb{H}^d)\), \(= \text{ the codomain restriction of } f'''\), where \(f''' (\mathbb{H}^d) \subseteq M_{2 \times 2 d} (\mathbb{C})\) is the topological subspace
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Conditions:
\(f'''^` \in \{\text{ the homeomorphisms }\}\)
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That means that the topology of \(\mathbb{H}^d\) is specified to make \(f'''^`\) homeomorphic.
2: Note
To be more specific, any \(S \subseteq \mathbb{H}^d\) is open if and only if \(f'''^` (S) \subseteq f''' (\mathbb{H}^d) \subseteq M_{2 \times 2 d} (\mathbb{C})\) is open.
As \(f'''^`\) is a bijection, it is indeed a topology.
An expected question is why \(\mathbb{H}^d\) is not made homeomorphic to \(\mathbb{R}^{4 d}\) by \(f'''^`: (a_1 + b_1 i + c_1 j + d_1 k, ..., a_d + b_d i + c_d j + d_d k) \mapsto (a_1, b_1, c_1, d_1, ..., a_d, b_d, c_d, d_d)\), which is of course possible, but is not so convenient, because it is not (at least directly) connected with convenient \(f\).
But in fact, \(\mathbb{H}^d\) is indeed homeomorphic to \(\mathbb{R}^{4 d}\) by that mapping, by the proposition that any quaternion Euclidean topological space is canonically homeomorphic to the corresponding Euclidean topological space.
