Definition:Jordan Curve/Also defined as

Jordan Curve: Also defined as

Some texts change the definition of the codomain of a Jordan curve from $\R^2$ to $X$, where $X$ is alternatively defined as:

• the complex plane $\C$
• a real Euclidean space $\R^n$
• a $T_2$ (Hausdorff) topological space $\struct { S, \tau_S }$

This is what $\mathsf{Pr} \infty \mathsf{fWiki}$ defines as an simple loop.

Some texts drop the condition that $\map f 0 = \map f 1$ and replace it with the condition:

$\map f t \ne \map f 1$ for all $t \in \openint 0 1$

which means they consider a Jordan arc to be a Jordan curve.

Some texts, especially those on topology, define a Jordan curve as a topological subspace $\struct{C, \tau_C}$ of $\R^2$ or $X$, where $\struct{C, \tau_C}$ is homeomorphic to the unit circle $\mathbb S^1$.

Jordan Curve Image Equals Set Homeomorphic to Circle shows the connection between the definition of Jordan curve as a path, and the definition as a topological space.

Sources

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• 2000: James R. Munkres: Topology (2nd ed.): $\S 66$
• 2014: Christopher Clapham and James Nicholson: The Concise Oxford Dictionary of Mathematics (5th ed.) ... (previous) ... (next): Jordan curve