Topology Glossary

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

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

Next, we come to embedding. It is important to understand this term because Boy's Surface is an immersion of the real projective plane embedded in 3-dimensional space. An embedding is the instance of one topological object, such as a manifold or graph, inside another topological object in such a way that certain properties are preserved. In topological spaces, an embedding specifically preserves open sets.

Euler Characteristic

The Euler Characteristic is calculated using a Triangulation, simply the division of a surface into triangles. However these triangles have the following restrictions: The intersection of any two triangles must be:

  • A single point that is the vertex of each triangle
  • A single edge that is a side of each of the triangles

The images below can be found on Cornell's website for mathematics. These are examples of triangulations:

Triangulation Example.jpg

Triangulation plays a major role in the Euler Characteristic, which is equal to the number of vertices minus the number of edges plus the number of triangles in the triangulation. This is shown algebraically as:

\boldsymbol{\chi}=v-e+f

where \boldsymbol{\chi} is the greek letter chi and where f stands for the number of triangles.

Fundamental Polygon

Homeomorphism

Immersion

Finally we come to our last major term, Immersion. I will first give you the mathematical definition according to WolframMathWorld:

A special nonsingular map from one manifold to another such that at every point in the domain of the map, the derivative is an injective linear transformation.

In topology, a map is a continuous function, meaning that each input has only one output. Below is an example of a mapping of continuity and non-continuity:

Continuous.jpeg Non-Continuous.jpeg

Continuity directly relates to Injective Linear Transformation such that for injective functions, there are always an equal number of outputs for the number of inputs (shown above). This can be visualized algebraically as well. For example, say you have a function such that f(x)=x^2 with inputs 3 and -3. When substituting these into the function, both result with 9. This function therefore is not continuous because it is not one-to-one, meaning that for each input there is not exactly one output.

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.

Real Projective Plane

The Real Projective Space is a modified Euclidean space (the typical x-y-z space) where every line in the projective space forms into a circle by meeting another point in the space. This is true for all line, even parallel lines. An example, would be on the road. Solid white lines indicating an emergency lane, meet at a point on the horizon, outline a possible visual interpertation of the real projective space. The projective space is constructed out of the many circles with an additional circle at infinity.

The Real Projective Plane (R'P2) is the 2 dimensional Real Projective Space. The Real Projective Plane has no edges, so the surface never intersects itself. The real projective plane cannot be shown in three space without it passes through itself somewhere, so, it is a four-dimensional figure. More importantly, the real projective plane is non-orientable.

Singularity

Topological Invariant

References