What is topology?

[Treadstone]

gonna take a class next semester on toplogy and i was wondering what it was. The course description was...less than descripted, damn mathmeticians

[5614]

ok, i have always associated topology with computers and networks.

 

a quick google define: search shows us this:

http://www.google.com/search?hl=en&q=define%3A+topology

mainly networks and computers.... as i though, here are the maths ones:

 

"The spatial relationships between connecting or adjacent coverage features (e.g., arcs, nodes, polygons, and points). For example, the topology of an arc includes its from- and to-nodes, and its left and right polygons. Topological relationships are built from simple elements into complex elements: points (simplest elements), arcs (sets of connected points), areas (sets of connected arcs), and routes (sets of sections, which are arcs or portions of arcs). Redundant data (coordinates) are eliminated because an arc may represent a linear feature, part of the boundary of an area feature, or both. Topology is useful in GIS because many spatial modeling operations don't require coordinates, only topological information. For example, to find an optimal path between two points requires a list of the arcs that connect to each other and the cost to traverse each arc in each direction. Coordinates are only needed for drawing the path after it is calculated."

 

"The study of how geometric objects are intrinsically connected to themselves. Since topologists are not concerned with the geometric measurements of objects, people often say that they study objects up to continuous deformation. But usually topologists consider spaces which have a topology (a qualitative shape or connectivity) but no predefined (quantitative) geometry. Knots and manifolds are typical examples of topological spaces. "

 

"an exact copy of the first one i quoted!"

 

"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 one-to-one correspondence that is continuous in both directions"

[Dave]

Topology is a little (read: lot) more complex than computing and networking. It takes the ideas in geometry and extrapolates them to looking at more complex systems, basically. I don't know a lot more than that myself tbh.

[MandrakeRoot]

I think the best way to describe topology is by making a contrast with metric spaces.

In metric spaces you study the propeties of a space using a notion of distance.

In topology you study spaces and their propeties using an imposed structure of "open sets".

Any metric space is particularly a topological (vector) space. Topological spaces are far more general and you can easily construct such spaces, wherein there exists sequences converging to every point of the space, something which is impossible in any metric space.

 

Mandrake

[Treadstone]

well it sounds pretty cool, though i only have a bit more of an understanding than i do now, lol....i'll post my findings while i'm taking the class

[Bernstein]

well it sounds pretty cool, though i only have a bit more of an understanding than i do now, lol....i'll post my findings while i'm taking the class

 

I looked it up in the dictionary and found some definitions:

 

topology n. pl. topologies

Topographic study of a given place, especially the history of a region as indicated by its topography.

Medicine. The anatomical structure of a specific area or part of the body.

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.

Computer Science. The arrangement in which the nodes of a LAN are connected to each other.

 

topology

n 1: topographic study of a given place (especially the history of place as indicated by its topography); "Greenland's topology has been shaped by the glaciers of the ice age" 2: 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 [syn: regional anatomy, topographic anatomy] 3: 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 one-to-one correspondence that is continuous in both directions [syn: analysis situs] 4: the configuration of a communication network [syn: network topology]

 

topology

1. <mathematics> The branch of mathematics dealing with

continuous transformations.

 

2. <networking> Which hosts are directly connected to which

other hosts in a network. Network layer processes need to

consider the current network topology to be able to route

packets to their final destination reliably and efficiently.

[Treadstone]

ah, i know a little about homomorphisms, so i think i know a little about what it is now...at least enough to tell people when they ask

[matt grime]

homEomorphisms, not homomorphisms

[Ophiolite]

Shape shifting!