# 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

In mathematics, the **Poincaré conjecture** (French, French pronunciation: [pwɛ̃kaʀe]) is a theorem about the characterization of the three-dimensional sphere among three-dimensional manifolds. It began as a popular, important conjecture, but is now considered a theorem to the satisfaction of the awarders of the Fields Medal. The claim concerns a space that locally looks like ordinary three-dimensional space but is connected, finite in size, and lacks any boundary (a closed 3-manifold). The Poincaré conjecture claims that if such a space has the additional property that each loop in the space can be continuously tightened to a point; then it is just a three-dimensional sphere. An analogous result has been known in higher dimensions for some time.

After nearly a century of effort by mathematicians, Grigori Perelman sketched a proof of the conjecture in a series of papers made available in 2002 and 2003. The proof followed the program of Richard S. Hamilton. Several high-profile teams of mathematicians have since verified the correctness of Perelman's proof.

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

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## 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.

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## 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 |
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