1DMannigfaltigkeit

1DMannigfaltigkeit, a term derived from German, translates to "one-dimensional manifold." In mathematics, a manifold is a topological space that locally resembles Euclidean space. A one-dimensional manifold is a space that, in the neighborhood of every point, looks like a line. These are the simplest types of manifolds.

Examples of one-dimensional manifolds include a circle and a line. A circle is a closed, bounded one-dimensional manifold, meaning it has no endpoints and is finite in extent. A line, on the other hand, is an open, unbounded one-dimensional manifold, extending infinitely in both directions. Other examples include the interval [0, 1] (a closed one-dimensional manifold with two endpoints) or the interval (0, 1) (an open one-dimensional manifold with no endpoints).

One-dimensional manifolds can be classified into two main categories: those that are connected and those that are not. A connected one-dimensional manifold is either homeomorphic to a circle (compact) or homeomorphic to the real line (non-compact). Disconnected one-dimensional manifolds are simply collections of disjoint copies of these connected examples. The study of manifolds is a fundamental area within differential geometry and topology, providing a framework for understanding many geometric objects.