In mathematics, topology (from the Greek words τόπος, 'place, location', and λόγος, 'study') is concerned with the properties of a geometric object that are preserved under continuous deformations, such as stretching, twisting, crumpling, and bending; that is, without closing holes, opening holes, tearing, gluing, or passing through itself.
A topological space is a set endowed with a structure, called a topology, which allows defining continuous deformation of subspaces, and, more generally, all kinds of continuity. Euclidean spaces, and, more generally, metric spaces are examples of a topological space, as any distance or metric defines a topology. The deformations that are considered in topology are homeomorphisms and homotopies. A property that is invariant under such deformations is a topological property. Basic examples of topological properties are: the dimension, which allows distinguishing between a line and a surface; compactness, which allows distinguishing between a line and a circle; connectedness, which allows distinguishing a circle from two non-intersecting circles.
The ideas underlying topology go back to Gottfried Leibniz, who in the 17th century envisioned the geometria situs and analysis situs. Leonhard Euler's Seven Bridges of Königsberg problem and polyhedron formula are arguably the field's first theorems. The term topology was introduced by Johann Benedict Listing in the 19th century, although it was not until the first decades of the 20th century that the idea of a topological space was developed.
In a quantum gravity discussion ("Chunkymorphism" thread) some issues of basic topology and measure theory came up. Might be fun to have a thread for such discussions.
for instance the statement was made, apparently concerning the real line (or perhaps more generally) that a countable set...
Hello everybody,
I am facing a Topology problem, and I hope you may be able to help me.
Let me try to describe my problem as clearly as I can: assume you have a set F of functions, such that any element f in F is a one-dimensional bounded and continuous function with common support S...
Let f be a real-valued function defined and continuous on the set of real numbers R. Which of the following must be true of the set S = {f(c): 0<c<1}?
I. S is a connected subset of R
II. S is an open subset of R
III. S is a bounded subset of R
The answer is I and III only. I understand...
a line segment (including its endpoints) is toplogically equivalent to a point.
consider S a nonempty totally ordered (ie a set with a relation <= such that <= is reflexive, transitive, x<=y and y<=x imply x=y, and every pair is comparable--i think that's what total order means anyway) set...
Greetings,
Having decided on a physics/math double major next year, I decided to get a head start this summer.
After tackling some classical mechanics, my next target is topology.
My problem is the following: I have been informed that there are two approaches to the subject, one...