Topology Glossary

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

Abstract (Manifold)

In geometry, we have many shapes with specific names, dimensions and properties. For instance, we are all familiar with a square, it has 2 dimensions, a width, a base, and other cool math properties. In topology, we have manifolds. A manifold is a broad definition of a shape. Manifolds are thought of as surfaces without any boundaries or edges.

Manifolds can be categorized by their dimensions. A one-dimensional manifold is just a one-dimensional shape or surface. This means each section of a one-dimensional manifold looks like a line. A two-dimensional manifold is just a two dimensional shape or surface. This means each section of a two dimensional manifold looks like a plane. In fact, a surface is a two dimensional manifold. Manifolds are the first step in understanding what type of surface Boy's Surface actually is.

An example of a manifold could be a tossed blanket. The tossed blanket is a shape. It has dimensions just like the typical square, even if we do not really think about it in that way. We don't have a particular name for this shape, but it is a shape nonetheless with many properties.

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

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 phenomenon.

Singularity

Topological Invariant

References