Definitions
from The American Heritage® Dictionary of the English Language, 5th Edition.
 noun Topographic study of a given place, especially the history of a region as indicated by its topography.
 noun Medicine The anatomical structure of a specific area or part of the body.
 noun The study of certain properties that do not change as geometric figures or spaces undergo continuous deformation. These properties include openness, nearness, connectedness, and continuity.
 noun The underlying structure that gives rise to such properties for a given figure or space.
 noun Computers The arrangement in which the nodes of a network are connected to each other.
from The Century Dictionary.
 noun The art or method of assisting the memory by associating the objects to be remembered with some place which is well known.
 noun A branch of geometry having reference to the modes of connection of lines and surfaces, but not to their shapes.
 noun The study and description of the localities in a particular district.
from the GNU version of the Collaborative International Dictionary of English.
 noun rare The art of, or method for, assisting the memory by associating the thing or subject to be remembered with some place.
 noun 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 .  noun configuration, especially in three dimensions;  used, e. g. of the configurations taken by macromolecules, such as superhelical DNA.
from Wiktionary, Creative Commons Attribution/ShareAlike License.
 noun mathematics A branch of mathematics studying those
properties of ageometric figure orsolid that are not changed bystretching ,bending and similarhomeomorphisms .  noun mathematics 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 arbitraryunions andfinite intersections .  noun medicine The
anatomical structure of part of the body.  noun computing The
arrangement ofnodes in acommunications network .  noun technology The properties of a particular technological embodiment that are not affected by differences in the physical layout or form of its application.
 noun The topographical study of geographic locations or given places in relation to its history.
 noun dated The art of, or method for, assisting the
memory by associating the thing or subject to be remembered with some place.
from WordNet 3.0 Copyright 2006 by Princeton University. All rights reserved.
 noun topographic study of a given place (especially the history of the place as indicated by its topography)
 noun the configuration of a communication network
 noun 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
 noun 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
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Examples

Network Topologies:  The term topology refers to way network is laid out, either physically or logically.
Recently Uploaded Slideshows prabhataits 2010

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.
November 2007 2007

After all, topology is affected by character distribution of outgroups.
Goodbye, my giant predatory, cursorial, flightless hoatzin Darren Naish 2006

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.
Arabic calligraphy in many forms Ray Girvan 2004

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.
Archive 20040801 Ray Girvan 2004

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) 2008

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 2010

Knot theory is a subspecialty of a field of mathematics known as topology, which is concerned with the study of spaces.
A math problem stumped experts for 50 years. This grad student from Maine solved it in days  The Boston Globe John WolfsonUpdated August 20, 2020, 12:32 p.m. 2020

Perturbations such as magnetic disturbances might slightly move the paths around without changing the qualitative nature of their linking, called their topology.
Physicists create longsought topological quantum states Davide Castelvecchi 2023
seanahan commented on the word topology
Most commonly heard (by me) as a type of mathematics.
February 29, 2008
ruzuzu commented on the word topology
One of the Century definitions ("The art or method of assisting the memory by associating the objects to be remembered with some place which is well known.") reminds me of a memory palace.
August 15, 2022