Definitions
from The American Heritage® Dictionary of the English Language, 4th Edition
 n. Topographic study of a given place, especially the history of a region as indicated by its topography.
 n. Medicine The anatomical structure of a specific area or part of the body.
 n. Mathematics The study of the properties of geometric figures or solids that are not changed by homeomorphisms, such as stretching or bending. Donuts and picture frames are topologically equivalent, for example.
 n. Computer Science The arrangement in which the nodes of a LAN are connected to each other.
from Wiktionary, Creative Commons Attribution/ShareAlike License
 n. A branch of mathematics studying those properties of a geometric figure or solid that are not changed by stretching, bending and similar homeomorphisms.
 n. A collection τ of subsets of a set X such that the empty set and X are both members of τ and τ is closed under arbitrary unions and finite intersections.
 n. The anatomical structure of part of the body.
 n. The arrangement of nodes in a communications network.
 n. The properties of a particular technological embodiment that are not affected by differences in the physical layout or form of its application.
 n. The topographical study of geographic locations or given places in relation to its history.
 n. The art of, or method for, assisting the memory by associating the thing or subject to be remembered with some place.
from the GNU version of the Collaborative International Dictionary of English
 n. The art of, or method for, assisting the memory by associating the thing or subject to be remembered with some place.
 n. a branch of mathematics which studies the properties of geometrical forms which retain their identity under certain transformations, such as stretching or twisting, which are homeomorphic. See also topologist.
 n. configuration, especially in three dimensions;  used, e. g. of the configurations taken by macromolecules, such as superhelical DNA.
from The Century Dictionary and Cyclopedia
 n. The study and description of the localities in a particular district.
 n. The art or method of assisting the memory by associating the objects to be remembered with some place which is well known.
 n. A branch of geometry having reference to the modes of connection of lines and surfaces, but not to their shapes.
from WordNet 3.0 Copyright 2006 by Princeton University. All rights reserved.
 n. topographic study of a given place (especially the history of the place as indicated by its topography)
 n. the configuration of a communication network
 n. the study of anatomy based on regions or divisions of the body and emphasizing the relations between various structures (muscles and nerves and arteries etc.) in that region
 n. the branch of pure mathematics that deals only with the properties of a figure X that hold for every figure into which X can be transformed with a onetoone correspondence that is continuous in both directions
Etymologies
Sorry, no etymologies found.
Examples

Network Topologies:  The term topology refers to way network is laid out, either physically or logically.

Algebraic topology is behind network theory which is used, among other things, to route circuits on CPUs.

A specialist in topology, he authored Grassmannians and Gauss Maps in PiecewiseLinear Topology (SpringerVerlag, 1987), but he was better known to the general public for his critiques of pseudoscience and obscurantism, including Higher Superstition: The Academic Left and Its Quarrels with Science (Johns Hopkins University Press, 1994), coauthored with Paul R.

The Journal of Topology publishes papers of high quality and significance in topology, geometry, and adjacent areas of mathematics.

After all, topology is affected by character distribution of outgroups.

As long as the basic topology is there, an extremely wide variation of shapes can be interpreted thus, due to written Arabic having multiple styles with vastly more flexible typography than Roman text.

a positivity but only as a topology, is it about history?
'The Abyss of the Past': Psychoanalysis in Schelling's Ages of the World (1815)

Ask yourself if that topology is congruent with Dawkins’s picture, or any textbook treatment of the same question.

EMU external integrity reads at maximum, with some warpages in topology.
365 tomorrows » 2010 » June : A New Free Flash Fiction SciFi Story Every Day

Instead, I took topology, which is sort of like geometry where the entire world is made of infinitely flexible rubber.
A debate centering around etymological and logical fallacies « Motivated Grammar
seanahan commented on the word topology
Most commonly heard (by me) as a type of mathematics.
February 29, 2008