Homeomorphic in the context of "Neoclassical compound"

In mathematics, a manifold is a topological space that locally resembles Euclidean space near each point. More precisely, an ${\displaystyle n}$-dimensional manifold, or ${\displaystyle n}$-manifold for short, is a topological space with the property that each point has a neighborhood that is homeomorphic to an open subset of ${\displaystyle n}$-dimensional Euclidean space.

One-dimensional manifolds include lines and circles, but not self-crossing curves such as a figure 8. Two-dimensional manifolds are also called surfaces. Examples include the plane, the sphere, and the torus, and also the Klein bottle and real projective plane.

In topology and related areas of mathematics, a topological property or topological invariant is a property of a topological space that is invariant under homeomorphisms. Alternatively, a topological property is a proper class of topological spaces which is closed under homeomorphisms. That is, a property of spaces is a topological property if whenever a space X possesses that property every space homeomorphic to X possesses that property. Informally, a topological property is a property of the space that can be expressed using open sets.

A common problem in topology is to decide whether two topological spaces are homeomorphic or not. To prove that two spaces are not homeomorphic, it is sufficient to find a topological property which is not shared by them.

Invariance of domain is a theorem in topology about homeomorphic subsets of Euclidean space ${\displaystyle \mathbb {R} ^{n}}$. It states:

The theorem and its proof are due to L. E. J. Brouwer, published in 1912. The proof uses tools of algebraic topology, notably the Brouwer fixed point theorem.

In mathematics, a solid torus is the topological space formed by sweeping a disk around a circle. It is homeomorphic to the Cartesian product ${\displaystyle S^{1}\times D^{2}}$ of the disk and the circle, endowed with the product topology.

A standard way to visualize a solid torus is as a toroid, embedded in 3-space. However, it should be distinguished from a torus, which has the same visual appearance: the torus is the two-dimensional space on the boundary of a toroid, while the solid torus includes also the compact interior space enclosed by the torus.