The Columbia Encyclopedia, 6th ed.


topology, branch of mathematics, formerly known as analysis situs, that studies patterns of geometric figures involving position and relative position without regard to size. Topology is sometimes referred to popularly as "rubber-sheet geometry" because a figure can be changed to that of an equivalent figure by bending, stretching, twisting, and the like, but not by tearing or cutting.

Branches of Topology

Topology may be roughly divided into point-set topology, which considers figures as sets of points having such properties as being open or closed, compact, connected, and so forth; combinatorial topology, which, in contrast to point-set topology, considers figures as combinations (complexes) of simple figures (simplexes) joined together in a regular manner; and algebraic topology, which makes extensive use of algebraic methods, particularly those of group theory. There is considerable overlap among these branches.

Continuous Transformations and Equivalent Figures

Topology is concerned with those properties of geometric figures that are invariant under continuous transformations. A continuous transformation, also called a topological transformation or homeomorphism, is a one-to-one correspondence between the points of one figure and the points of another figure such that points that are arbitrarily close on one figure are transformed into points that are also arbitrarily close on the other figure. Figures that are related in this way are said to be topologically equivalent. If a figure is transformed into an equivalent figure by bending, stretching, etc., the change is a special type of topological transformation called a continuous deformation. Two figures (e.g, certain types of knots) may be topologically equivalent, however, without being changeable into one another by a continuous deformation.

It is intuitively evident that all simple closed curves in the plane and all polygons are topologically equivalent to a circle; similarly, all closed cylinders, cones, convex polyhedra, and other simple closed surfaces are equivalent to a sphere. On the other hand, a closed surface such as a torus (doughnut) is not equivalent to a sphere, since no amount of bending or stretching will make it into a sphere, nor is a surface with a boundary equivalent to a sphere, e.g., a cylinder with an open top, which may be stretched into a disk (a circle plus its interior).

Topological Properties

There are various properties of a figure, in general, and of a surface such as a sphere, torus, or disk, in particular, that may be used to distinguish between such figures topologically. One property is the number of boundaries the surface has, if any. Another property is orientability; a surface is orientable if a circle drawn on it with a given orientation (clockwise or counterclockwise) always, if moved around the surface, returns to its original position with the same orientation. A sphere and a torus are both orientable, but a Möbius strip (a one-sided surface made by twisting a strip of paper and joining the ends so that opposite edges correspond) is a nonorientable surface, since an oriented circle moved around the strip will return to its original position with its orientation reversed (see Möbius, Augustus Ferdinand).

Another topological property of a surface is its Euler-Poincaré characteristic, a number which can be calculated from any polyhedral decomposition of the surface. If V is the number of points (vertices) in the decomposition, E is the number of line segments (edges), and F is the number of regions (faces), then the characteristic is given by Χ=V-E+F and is the same for all possible polyhedral decomposition of the given surface. For a sphere, Χ=2, and the formula is identical with Euler's formula for the vertices, edges, and faces of a spherical polyhedron, to which the sphere is topologically equivalent. For a torus, Χ=0. The Euler-Poincaré characteristic for an orientable surface is Χ=2-2p, where p is called the genus of the surface. Any orientable closed surface is topologically equivalent to a sphere with p handles attached to it; e.g., the torus, having Χ=0, is of genus 1 and is equivalent to a sphere with one handle, and a double torus (two-hole doughnut), equivalent to a sphere with two handles, is of genus 2 and has Χ=-2. For a nonorientable surface, Χ=2-q, where q is the number of cross-caps that must be added to a sphere to make it equivalent to the surface. (A cross-cap is a cap with a twist like a Möbius strip in it.)

Closely related to the Euler-Poincaré characteristic is the connectivity number of a surface, which is equal to the largest number of closed cuts (or cuts connecting points on boundaries or on previous cuts) that can be made on the surface without separating it into two or more parts. The connectivity number is equal to 3-Χ for a closed surface and to 2-Χ for a surface with boundaries (e.g., a disk). A surface with a connectivity number of 1, 2, or 3 is said to be simply connected, doubly connected, or triply connected, respectively, and similarly for more complex surfaces; a sphere is simply connected, while a torus is triply connected. Thus, any surface can be classified by its boundary curves (if any), its orientability, and its Euler-Poincaré characteristic or connectivity number; and any surface is topologically equivalent to a sphere with an appropriate number of handles, cross-caps, or holes. A surface is a simple example of a topological space, the basic entity studied in topology.

Different types of topological spaces are defined according to axioms satisfied by the sets of points that constitute the space. Especially important are topological spaces for which a distance function is defined for every pair of points in the space; such spaces are called metric spaces. A full treatment of the properties of topological spaces of arbitrary dimension requires various concepts of an advanced nature, e.g., homology theory, and is beyond the scope of a general article. The most important spaces, manifolds, are those which are locally equivalent to the Euclidean space of the same dimension. The fundamental problem of classifying manifolds was classically solved for dimensions 1 and 2, and largely clarified in dimensions 5 or more during the past 30 years. Dimensions 3 and 4 are now areas of vigorous research, stimulated in part by ideas from physics. The theory of knots plays an important role in dimension 3, and has revealed surprising connections with physics and application to biology.

Notes for this article

Add a new note
If you are trying to select text to create highlights or citations, remember that you must now click or tap on the first word, and then click or tap on the last word.
One moment ...
Default project is now your active project.
Project items
Cite this article

Cited article

Citations are available only to our active members.
Buy instant access to cite pages or passages in MLA 8, MLA 7, APA and Chicago citation styles.

(Einhorn, 1992, p. 25)

(Einhorn 25)

(Einhorn 25)

1. Lois J. Einhorn, Abraham Lincoln, the Orator: Penetrating the Lincoln Legend (Westport, CT: Greenwood Press, 1992), 25,

Note: primary sources have slightly different requirements for citation. Please see these guidelines for more information.

Cited article



Text size Smaller Larger Reset View mode
Search within

Search within this article

Look up

Look up a word

  • Dictionary
  • Thesaurus
Please submit a word or phrase above.
Print this page

Print this page

Why can't I print more than one page at a time?

Full screen
Items saved from this article
  • Highlights & Notes
  • Citations
Some of your highlights are legacy items.

Highlights saved before July 30, 2012 will not be displayed on their respective source pages.

You can easily re-create the highlights by opening the book page or article, selecting the text, and clicking “Highlight.”

matching results for page

    Questia reader help

    How to highlight and cite specific passages

    1. Click or tap the first word you want to select.
    2. Click or tap the last word you want to select, and you’ll see everything in between get selected.
    3. You’ll then get a menu of options like creating a highlight or a citation from that passage of text.

    OK, got it!

    Cited passage

    Citations are available only to our active members.
    Buy instant access to cite pages or passages in MLA 8, MLA 7, APA and Chicago citation styles.

    "Portraying himself as an honest, ordinary person helped Lincoln identify with his audiences." (Einhorn, 1992, p. 25).

    "Portraying himself as an honest, ordinary person helped Lincoln identify with his audiences." (Einhorn 25)

    "Portraying himself as an honest, ordinary person helped Lincoln identify with his audiences." (Einhorn 25)

    "Portraying himself as an honest, ordinary person helped Lincoln identify with his audiences."1

    1. Lois J. Einhorn, Abraham Lincoln, the Orator: Penetrating the Lincoln Legend (Westport, CT: Greenwood Press, 1992), 25,

    Cited passage

    Thanks for trying Questia!

    Please continue trying out our research tools, but please note, full functionality is available only to our active members.

    Your work will be lost once you leave this Web page.

    Buy instant access to save your work.

    Already a member? Log in now.

    Search by... Author
    Show... All Results Primary Sources Peer-reviewed


    An unknown error has occurred. Please click the button below to reload the page. If the problem persists, please try again in a little while.