nLab Jordan curve

Jordan curves

Jordan curves

Definition

A continuous simple closed curve, or Jordan curve, in a topological space (or convergence space, locale, etc) XX is the image of a continuous injection to XX from the unit circle S 1S^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 1B^1 instead of S 1S^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 XX is the image of a continuous injection to XX from the unit sphere S 2S^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 XX has dimension nn (in some understood sense), then a Jordan hypersurface in XX is the image of a continuous injection to XX from S n1S^{n-1}.

In cohesive homotopy type theory

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

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

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

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