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The wikipedia article says, 'The definition of a topological space relies only upon
set theory and is the most general notion of a mathematical "space" that allows for the definition of concepts such as continuity, connectedness, and convergence.'
These are abstract spaces where distance is, in some sense, ignored. When Euler considered the 'seven bridges of Koenigsberg problem', for instance, he appreciated that he was ignoring the distances between the bridges and was considering only how they were connected--so that someone could traverse each of them just once. Since that time, of course, the idea of a topological space has permeated many areas of mathematics.
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What is a Betti number?
A Betti number is a number associated to each topological space and dimension, giving an approximate number of holes of that dimension in that space.
What does the interior of a subspace within a topological space mean?
Let B, t be a topological space and let C ⊂ B. The interior of C, written Co is the union of all of the open sets within C. This can be expressed using set theory notation asCo = ∪{P Є t | P ⊂ C}.See related links for more information.
What does it mean for a subset to be open?
A subset C of set B associated with the topological space B, D is said to be open if at any point c Є C there exists a positive number p such that the D-p neighborhood N(x, p) ⊂ C.See related links for more information and definitions.
What is topology and example?
Topology is the study of objects (often surfaces in 3D) where details like position, shape and curvature are unimportant. Two topological objects are considered equivalent if they can be stretched to look like one another. An example of two different topological objects are a sphere and a doughnut shape (torus); the sphere cannot be stretched to look like the torus because it doesn't have a hole and the torus does.
What is the difference between set and space in mathematics?
A space is a set with structure. A number of different kinds of mathematical structures (or topologies) exist, including metrics, norms, and inner products. Sets paired with each of these result in a different kinds of spaces, each with a host of interesting properties. Examples of metric spaces include 2-dimensional Euclidean space (as in the surface of a flat sheet of paper), 3-dimensional Euclidean space (a simplification of the world we live in), the Minkowski space (our 3-D world subjected to Einstein's special relativity), and elliptic geometry (which can be used to measure the distances between locations on the surface of the Earth). There also exist topological spaces that are not metric spaces, i.e. spaces that do not have a strict notion of distance between their points. The same set may be paired with different topologies (or different metrics, if applicable), and each of these pairings should be thought of as forming distinct topological (or metric) spaces. There are no spaces that are not sets. On the other hand, any set not paired with a topology is not a space. One can, however, pair any set with the trivial metric d(x,y) = { 0 iff x = y, 1 otherwise } to arrive at a trivial topology. Disregarding this, it is easy to imagine sets that are not spaces, such as for example the set of all automobile models with model year 2013.
