# Portal:Topology

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## 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, a **metric space** is a set where a notion of distance (called a metric) between elements of the set is defined.

The metric space which most closely corresponds to our intuitive understanding of space is the 3-dimensional Euclidean space. In fact, the notion of "metric" is a generalization of the Euclidean metric arising from the four long known properties of the Euclidean distance. The Euclidean metric defines the distance between two points as the length of the straight line connecting them.

The geometric properties of the space depends on the metric chosen, and by using a different metric we can construct interesting non-Euclidean geometries such as those used in the theory of general relativity.

A metric space also induces topological properties like open and closed sets which leads to the study of even more abstract topological spaces.

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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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