Topology Glossary

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This page provides an introduction to many topological terms and concepts.

Abstract (Manifold)

Boundary

In Topology, the term boundary refers to the edge or edges of a figure. The sphere has no boundary, while the circle has one, and the annulus has two.

The edge of a 1 dimensional object is a corner, while that of a 2 dimensional object is a side, like the sides of a square. The edges of a 3 dimensional object is what is commonly called a surface. The boundary of a table is the table-top, sides, and bottom.

Boundary is not the same as bounded.

Bounded

A bounded entity is one in which there is a limit to how far away two points can be from each other. In a certain sense, it means that the shape does not extend infinity in any direction.

Bounded is not the same as boundary.

Closed

Closed sets contain all of their boundaries. If a set contains no boundaries, then this is satisfied by default. Thus, a topological set, or entity, is closed if it contains all points which are part of the figure. Hence, if there are edges on a shape, then the figure must include these edges in order to be closed.


Compact

A compact figure is, in essence, one that is both closed and bounded.

Dimension

Please follow this link: Dimensions.

Embedding

Fundamental Polygon

Homeomorphism

Immersion

Manifold

Simply put, a manifold is a shape without edges. Manifolds are classified by their dimensionality; for instance, a line segment is 1 dimensional, so is called a 1 manifold. A cube is a 3 dimensional manifold, or 3 manifold. A manifold with n dimensions is an n manifold. There are manifolds with any and every number of dimensions.

The main characteristic of a manifold is that a small, n - 1 dimensional ball can be drawn around any point on the manifold without hitting a boundary. In effect, this means that manifolds have no boundaries, or edges.

Surface

A surface is a 2 dimensional manifold.

Map

Orientability

Orientability is a property of manifolds that describes whether directions, or orientations, can be consistently defined in the surface. Directions can be consistently defined on an orientable manifold, but not on a non-orientable manifold.

In the case of surfaces, a 2 mandifold is orientable if every closed path preserves the orientation of an entity that travels along it. A 2 mandifold is non-orientable if there exists at least one closed path that reverses the orientation of something traveling along it[1].

For instance, say that rotation in the counter-clockwise direction is considered to be positive, and clockwise, negative. If an object spinning counter-clockwise moves along a closed path in an orientable surface, it will return to its starting point spinning counter-clockwise. Yet, if the surface is non-orientable, then it will twist in such a way that, on certain paths, the object will return spinning clockwise, though the motion of its rotation never changed. Equivalently, in non-orientable surfaces, an object within the surface can travel along a path that will lead it back to its start point, but with its right and left sides flipped.

A good demonstration of non-orientability on the Mobius Strip is visible on this page from Plus Magazine. When viewing this, remember that the face is within the surface, not merely pasted on one of the sides.

Orientability and non-orientability are most often discussed in relation to surfaces. Nevertheless, the properties are descriptive of higher dimensional manifolds as well. One can consider a 3 dimensional space where certain paths would lead back to their start points, but flip the right and left sides of things that traveled along them.

Here is a more technical explanation: In a non-orientable manifold, there exists at least one path around the manifold such that, if we take a set of basis vectors for the manifold and move them along this path, they will arrive at their starting point with the following result. The determinant of the matrix composed of these basis vectors will have the opposite sign (positive/ negative) when it arrives back at the starting point as it did when it left.

Not all image pages will discuss in depth how non-orientability arises for a particular surface; the phenomenon is discussed, however, for the Mobius Strip and Real Projective Plane, with images to explain the phenominon.

Topological Invariant

References

  1. Massey, William. (1991). A Basic Course in Algebraic Topology (Graduate Texts in Mathematics). New York: Springer-Verlag.