In topology and related areas of mathematics a topological property or topological invariant is a property of a topological space which is invariant under homeomorphisms. That is given two topological spaces X and Y and a homeomorphism f between them, a topological property for a subset A of X holds if and only if it holds for f(A). Topology (Greek topos, place and logos, study) is a branch of mathematics concerned with the study of topological spaces. ... Wikibooks Wikiversity has more about this subject: School of Mathematics Wikiquote has a collection of quotations by or about: Mathematics Look up Mathematics in Wiktionary, the free dictionary Wikimedia Commons has more media related to: Mathematics Bogomolny, Alexander: Interactive Mathematics Miscellany and Puzzles. ... Topological spaces are structures that allow one to formalize concepts such as convergence, connectedness and continuity. ... An invariant in mathematics is something that does not change under a set of transformations. ... This word should not be confused with homomorphism. ... In mathematics, philosophy, logic and technical fields that depend on them, iff is used as an abbreviation for if and only if. It is often, not always, written italicized: iff. ...
A common problem in topology is to decide whether two topological spaces are homeomorphic or not. To prove that two spaces are not homeomorphic it is sufficient to find a topological property which is not shared by them. This word should not be confused with homomorphism. ...
Topologicalproperties may be classified by their behaviour with respect to mappings.
A property of a topological space is called continuous if it is the case that, whenever a space has this property, the images of this space under all continuous mapping also have the same property.
This is version 11 of classification of topologicalproperties according to behaviour under mapping, born on 2004-09-24, modified 2006-12-21.
The exact shapes of the pieces constitute a geometric property of the space, and the study of these shapes is in the domain of differential geometry, but the fact that the space has three separate pieces is a qualitative or topologicalproperty.
There are various properties of a figure, in general, and of a surface such as a sphere, torus, or disk, in particular, that may be used to distinguish between such figures topologically.
A traditional joke is that a topologist can't tell the coffee mug she is drinking out of from the doughnut she is eating, since a sufficiently pliable donut could be reshaped to the form of a coffee cup by creating a dimple and progressively enlarging it, while shrinking the hole into a handle.
Feeds |
Contact