Discrete space

In the mathematical branch of topology, a topological space is discrete if all points are isolated, ie if there are no other points in a sufficiently small neighborhood of the point.

Definition

It was a lot. Then the discrete topology on the topology under which all subsets of are open. A space which supports the discrete topology is, discrete.

In other words, wearing just the power set as topology.

Subsets of topological spaces are called discrete if it is discrete with the induced topology. This is equivalent to saying that at each point is a neighborhood of which contains a single point of, ie.

Category Theoretical background

Category theory it would be consistent to denote the discrete topology as free topology. To this end, consider the functor from the category of all sets (with all lot of pictures as morphisms ) to the category of all topological spaces (with all continuous maps as morphisms ), which each quantity assigns the discrete topological space and lots of illustration the same mapping between the associated discrete spaces. This functor is now linksadjungiert for forgetful. Usually, the images of sets under such functors are, however, referred to as free constructions, for example, free groups, free abelian groups, free modules. Similarly, the indiscrete topology is a functor rechtsadjungiert to the above forgetful. That is the indiscrete topology is the dual notion of discrete topology.

Properties

  • Continuous maps from a topological space into a discrete topological space is locally constant.
  • Lots of picture from a discrete topological space in an arbitrary topological space is continuous.
  • A topological space is discrete then, if for each point the set is open.
  • Discrete spaces are always Hausdorff. Then you have come to be compact if it contains only finitely many points.
  • Discrete spaces are locally compact.
  • The Cartesian product of a finite number of discrete topological spaces is discretely again.
  • The discrete topology is the discrete metric
  • Discrete spaces are totally disconnected: any subspace with at least two elements is incoherent, ie decomposes into two disjoint open sets.
  • Discrete spaces are 0- dimensional, both in regard to the small and large inductive dimension as well as with respect to the Lebesgue covering dimension.
Mathematics Isolated point Subspace topology Trivial topology Locally constant function
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