769: Curve on Topological Space

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definition of curve on topological space

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Topics

About: topological space

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The table of contents of this article

• Starting Context
• Target Context
• Orientation
• Main Body
• 1: Structured Description
• 2: Natural Language Description
• 3: Note

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Starting Context

• The reader knows a definition of continuous map.

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Target Context

• The reader will have a definition of curve on topological space.

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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.

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Main Body

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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 $\mathbb{R}$
$\ast \gamma$: $:J\to T$, $\in \{\text{ the continuous maps }\}$
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Conditions:
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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, $\gamma :J\to T$

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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 $C^{\mathrm{\infty }}$ manifold with boundary, because a curve that starts at a boundary point may need to have a half-closed interval.

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References

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