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What is topological space?

Updated: 9/23/2023
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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.

Related questions

Is a point is a zero dimensional?

In mathematics, a zero-dimensional topological space is a topological space that ... any point in the space is contained in exactly one open set of this refinement.


What notation is used to symbolize a topological space?

A topological space is simply a set, B, with topology t (see the related link for a definition), and is often denoted as B, t which is similar to how a metric space is often denoted; B, D.


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.


Is it proven that algenraic functions turn dyslexic topological polynomials on their head in Non-Non-Euclidean Riemannian after-space?

no


What is material uniformity?

A family of subsets of the direct product of a topological space with itself that is used to derive a uniform topology for the space. Also known as uniform structure.


What is the definition of coaser?

-adjective Mathematics. of or pertaining to a topology on a topological space whose open sets are included among the open sets of a second specified topology on the space.


What has the author Maria Fragoulopoulou written?

Maria Fragoulopoulou has written: 'Topological algebras with involution' -- subject(s): Topological algebras 'An introduction of the representation theory of topological *-algebras' -- subject(s): Topological algebras, Representations of algebras


Is Z with the discrete topology a compact topological space?

A discrete topology on the integers, Z, is defined by letting every subset of Z be open If that is true then Z is a discrete topological space and it is equipped with a discrete topology. Now is it compact? We know that a discrete space is compact if and only if it is finite. Clearly Z is not finite, so the answer is no. If you picked a finite field such a Z7 ( integers mod 7) then the answer would be yes.


What has the author R Lowen written?

R. Lowen has written: 'On the existence of natural non-topological, fuzzy topological spaces' -- subject(s): Topological spaces, Fuzzy sets


What has the author Eduard Cech written?

Eduard Cech has written: 'Point sets' -- subject(s): Set theory, Topological spaces 'Topological spaces' -- subject(s): Topological spaces


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 is up with that?

depends on what that means... but to answer you question... i don't know. algenraic functions turn dyslexic topological polynomials on their head in Non-Non-Euclidean Riemannian after-space