# 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 labeled graph on 6 vertices and 7 edges |

**graph**is a set of objects called

*points*,

*nodes*, or

*vertices*connected by links called

*lines*or

*edges*. In a proper graph, which is by default

*undirected*, a line from point

*A*to point

*B*is considered to be the same thing as a line from point

*B*to point

*A*. In a

*digraph*, short for

*directed graph*, the two directions are counted as being distinct

*arcs*or

*directed edges*. Typically, a graph is depicted in diagrammatic form as a set of dots (for the points, vertices, or nodes), joined by curves (for the lines or edges).

Applications of graph theory are generally concerned with labeled graphs and various specializations of these. Many problems of practical interest can be represented by graphs. The link structure of a website could be represented by a directed graph: the vertices are the web pages available at the website and a directed edge from page *A* to page *B* exists if and only if *A* contains a link to *B*. A graph structure can be extended by assigning a numerical value (known as a "weight") to each edge of the graph. For example, if a graph represents a road network, the weights could represent the length of each road. A digraph with weighted edges in the context of graph theory is called a network. Networks have many uses in the practical side of graph theory, network analysis (for example, to model and analyze traffic networks).

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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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