Jordan curves

# Jordan curves

## Definition

A continuous simple closed curve, or Jordan curve, in a topological space (or convergence space, locale, etc) $X$ is the image of a continuous injection to $X$ from the unit circle $S^1$. (This map itself is a continuous parametrization? of the curve.)

The word ‘continuous’ is generally assumed, so that one speaks simply of a simple closed curve. (If ‘closed’ is removed, then the domain is taken to be the unit interval $B^1$ instead of $S^1$. If ‘simple’ is removed, then the map is no longer assumed injective. If the image alone is not sufficient data, then the word ‘parametrized’ may be added to indicate the map itself while thinking of the map as its image.)

Similarly, a Jordan surface in $X$ is the image of a continuous injection to $X$ from the unit sphere $S^2$. This can be generalized to higher-dimensional spheres or other domains, so long as there is an appropriate term to use in place of ‘curve’ and ‘surface’. In particular, if $X$ has dimension $n$ (in some understood sense), then a Jordan hypersurface in $X$ is the image of a continuous injection to $X$ from $S^{n-1}$.

### In cohesive homotopy type theory

In cohesive homotopy type theory, let the continuum line object $\mathbb{A}$ be a commutative ring such that the shape of $\mathbb{A}$ is contractible $\mathrm{isContr}(\esh(\mathbb{A}))$. The unit circle is then defined to be the dependent sum type

$\mathbb{S}^1 \coloneqq \sum_{x:\mathbb{A}} \sum_{y:\mathbb{A}} x^2 + y^2 = 1$

A Jordan curve in a cohesive type $X$ is the image $J \equiv \mathrm{im}(i)$ of an embedding $i:\mathbb{S}^1 \hookrightarrow X$ from the unit circle $\mathbb{S}^1$ into $X$.

Commonly seen examples include in Euclidean geometry where $\mathbb{A}$ is the Dedekind real numbers $\mathbb{R}$, and algebraic geometry where $\mathbb{A}$ is the affine line $\mathbb{A}^1$.

## References

Last revised on October 23, 2022 at 03:09:39. See the history of this page for a list of all contributions to it.