# Portal:Topology

## Topology

**Topology** (Greek *topos*, "place," and *logos*, "study") is a branch of mathematics that is an extension of geometry. Topology begins with a consideration of the nature of space, investigating both its fine structure and its global structure. Topology builds on set theory, considering both sets of points and families of sets.

The word *topology* is used both for the area of study and for a family of sets with certain properties described below that are used to define a topological space. Of particular importance in the study of topology are functions or maps that are *homeomorphisms*. Informally, these functions can be thought of as those that stretch space without tearing it apart or sticking distinct parts together.

When the discipline was first properly founded, toward the end of the 19th century, it was called *geometria situs* (Latin *geometry of place*) and *analysis situs* (Latin *analysis of place*). From around 1925 to 1975 it was an important growth area within mathematics.

Topology is a large branch of mathematics that includes many **subfields**. The most basic division within topology is **point-set topology**, which investigates such concepts as compactness, connectedness, and countability; **algebraic topology**, which investigates such concepts as homotopy and homology; and **geometric topology**, which studies manifolds and their embeddings, including knot theory.

## Selected article

A 3-D depiction of a trefoil knot |

**Knot theory** is the mathematical branch of topology that studies mathematical knots, which are defined as embeddings of a circle in 3-dimensional Euclidean space, **R**^{3}. This is basically equivalent to a conventional knotted string with the ends joined together to prevent it from becoming undone. Two mathematical knots are equivalent if one can be transformed into the other via a deformation of **R**^{3} upon itself. These transformations correspond to manipulations of a knotted string that do not involve cutting the string or passing the string through itself.

Knots can be described in various ways. Given a method of description, however, there may be more than one description that represents the same knot. Therefore, a fundamental problem in knot theory is determining when two descriptions represent the same knot. One way of distinguishing knots is by using a *knot invariant*, a "quantity" which remains the same even with different descriptions of a knot.

...Archive |
Image credit: User:Jtico | Read more... |

## WikiProjects

The **Mathematics WikiProject** is the center for mathematics-related editing on Wikipedia. Join the discussion on the project's **talk page**.

**Project pages**

**Essays**

**Subprojects**

**Related projects**

## Selected picture

It is often suggested that a topologist cannot tell the difference between a coffee cup and a doughnut. This is because these objects when thought of as topological spaces are homeomorphic. The above picture depicts a continuous deformation of a coffee cup into a doughnut such that at each stage the object is homeomorphic to the original.

...Archive |
Read more... |

## Did you know?

- ...that it is possible to turn a sphere inside out without tears or creases? (The sphere is allowed to pass through itself).
- ...that the Klein bottle gives a two-fold covering space of itself?
- ...that outstanding mathematician Grigori Perelman was offered a Fields Medal in 2006, in part for his proof of the Poincaré conjecture, which he declined?
- ...that you cannot knot strings in 4-dimensions? You can, however, knot 2-dimensional surfaces like spheres.

## Categories

*Topics in Topology*

Main articles | Key concepts | Algebraic topology | Geometric topology |
---|---|---|---|

## Related portals

Algebra |
Analysis |
Categorytheory |
Computerscience |
Cryptography |
Discretemathematics |
Geometry |

Logic |
Mathematics |
Numbertheory |
Physics |
Science |
Set theory |
Statistics |
Topology |

**What are portals?****List of portals**

## Wikimedia

**What are portals?****List of portals**