definition of Euclidean topology
Topics
About: topological space
The table of contents of this article
- Starting Context
- Target Context
- Orientation
- Main Body
- 1: Structured Description
- 2: Natural Language Description
- 3: Note
Starting Context
- The reader knows a definition of topology induced by metric.
Target Context
- The reader will have a definition of Euclidean topology.
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:
\( \mathbb{R}^d\): with the Euclidean metric
\(*O\): \(= \text{ the topology induced by the Euclidean metric }\)
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Conditions:
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2: Natural Language Description
For the Euclidean set, \(\mathbb{R}^d\), the topology induced by the Euclidean metric
3: Note
The Euclidean topological space, \(\mathbb{R}^d\), does not need to really have the Euclidean metric: the metric is used just in order to define the topology and the metric can be forgotten after that if one likes so. In fact, the topology can be defined without the metric, but this definition uses the metric in order to use the fact that the topology induced by any metric is indeed a topology.
