definition of curve on topological space
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 continuous map.
Target Context
- The reader will have a definition of curve on 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:
\( T\): \(\in \{ \text{ the topological spaces } \}\)
\( \mathbb{R}\): \(= \text{ the Euclidean topological space } \)
\( J\): \(\in \{\text{ the intervals on } \mathbb{R}\}\), as the topological subspace of \(\mathbb{R}\)
\(*\gamma\): \(: J \to T\), \(\in \{\text{ the continuous maps }\}\)
//
Conditions:
//
2: Natural Language Description
For any topological space, \(T\), the Euclidean topological space, \(\mathbb{R}\), and any interval on \(\mathbb{R}\), \(J \subseteq \mathbb{R}\), as the topological subspace of \(\mathbb{R}\), any continuous map, \(\gamma: J \to T\)
3: Note
\(J\) does not need to be any open interval: \(J\) can be an open interval, \((t_0, t_1)\), a closed interval, \([t_0, t_1]\), a lower-open-upper-closed interval, \((t_0, t_1]\), or a lower-closed-upper-open interval, \([t_0, t_1)\).
Not demanding \(J\) to be any open interval is important for being used on a \(C^\infty\) manifold with boundary, because a curve that starts at a boundary point may need to have a half-closed interval.
