Quotient space (topology)

The quotient topology (also called topology identification ) is a term from the mathematical branch of topology. Intuitively, this topology if you " stick together " points, ie identified two distinct points as previously and the same point. Such points are defined by equivalence relations. This generally happens with new topological spaces from existing dissipate. For a generalization of this construction, the article compare the final topology.

Definition

It is a topological space and a surjective map of sets. Then the quotient topology induced by on the one in which a subset if and only open when the archetype is open.

Properties

  • The quotient topology on the finest topology for which the image is continuous.
  • Provides you with the quotient topology, so is a quotient map: Is another topological space and a mapping of the underlying quantities, so is continuous if is continuous ( universal property of the quotient topology):

Important special cases

  • If an equivalence relation on a topological space, so you know the set of equivalence classes usually without further mention with the induced from the canonical quotient topology mapping.
  • In particular, is a topological group and a subgroup of, so it provides the homogeneous spaces and with the quotient topology.
  • Beating up a subspace to a point: If a topological space and a subset of so called the set of equivalence classes with respect to the equivalence relation, called at the two points equivalent if they are equal or both are in. The map is injective outside, and the image of is a single point.

Note the likelihood of confusion in the notation.

Examples

  • It is the unit interval and the unit circle. Then which is the figure
  • Is the unit interval and so is the homeomorphic by striking together of resulting to a point on the space circle. This is essentially the same as the first example, but there the target quantity and the figure had already given explicitly, here it was only through the implicit during impact equivalence relation.
  • The homogeneous space is also homeomorphic to the circle
  • In contrast, the space that you get when you beat up the subset of a point, clearly spoken of infinitely many circles that were glued together at a point.
Pointed space
407238
de