Example of a differentiable curve or 1-manifold , which is also closed, that is, it is the image of an injective mapping :${\displaystyle S^{1}\to \mathbb {R} ^{3}}$. This curve is a knot called the shamrock knot .

In topology a 1- manifold is a topological space of dimension one.

For example, the number line [ 1 ] and the circumference [ 2 ] are 1-manifolds without boundary while the intervals ( bounded ) are 1-manifolds with boundary.

It is also true that the trajectories (not necessarily differentiable) and that they do not intersect themselves, are topological 1-dimensional varieties .

## Classification

From the topological point of view we have - for one- connected manifolds - the following homeomorphic types :

• to the number line : infinitely long sets (bi-laterally) and without boundary .
• to the ray : infinitely long sets (uni-laterally) and with a border of a single point.
• at intervals : infinite but bounded sets, with two boundaries of two disjoint points.
• to the circle : infinite, bounded and borderless sets.

For the disconnected they take any of the types above to find the appropriate combination.

## Notes and references

1. ie the real numbers .
2. In English: circle, cercle, Kreis .