Discrete topology

In the mathematical sub-area of topology , a topological space is discrete if all points are isolated , ie if there are no further points in a sufficiently small area around the point.

definition

It is a lot . Then the discrete topology is the topology under which all subsets of are open. A room that bears the discrete topology is called discrete . ${\displaystyle X}$ ${\displaystyle X}$ ${\displaystyle X}$

That means, just carries the power set as topology . ${\displaystyle X}$ ${\displaystyle {\mathcal {P}}(X)}$

Subsets of topological spaces are called discrete if they are discrete with the subspace topology . This is equivalent to the fact that for every point there is a neighborhood of which contains as the only point of , i.e. H. . ${\displaystyle Y}$ ${\displaystyle X}$ ${\displaystyle y\in Y}$ ${\displaystyle U\subseteq X}$ ${\displaystyle y}$ ${\displaystyle y}$ ${\displaystyle Y}$ ${\displaystyle U\cap Y=\{y\}}$

properties

• A topological space is discrete if and only if the set is open for every point . ${\displaystyle X}$ ${\displaystyle x\in X}$ ${\displaystyle \{x\}}$
• In a discrete topological space , the environment filter of each point is the set of all subsets with He is an ultrafilter . ${\displaystyle X}$ ${\displaystyle x\in X}$ ${\displaystyle U\subseteq X}$ ${\displaystyle x\in U.}$
• In a discrete topological space the filter is convergent if and only if it is the environment filter of a point . This point is then the limit point of the filter ${\displaystyle X}$ ${\displaystyle {\mathcal {F}}}$ ${\displaystyle x\in X}$ ${\displaystyle x}$ ${\displaystyle {\mathcal {F}}.}$
• A sequence from a discrete topological space converges if and only if it becomes constant from a certain term in the sequence (in other words, it is stationary ). ${\displaystyle \left(x_{n}\right)_{n\in \mathbb {N} }}$
• Discrete rooms are always Hausdorff-like . They are compact if and only if they contain a finite number of points.
• Discrete spaces are locally compact .
• The Cartesian product of a finite number of discrete topological spaces is again discrete.
• Discrete spaces are totally disconnected: Any subspace with at least two elements is disconnected, i.e. it breaks down into (at least) two disjoint open sets.
• Discrete spaces are 0-dimensional, both with regard to the small and large inductive dimension and with regard to the Lebesgue coverage dimension .
• Every mapping from a discrete topological space into any topological space is continuous. ${\displaystyle X}$ ${\displaystyle Y}$
• A continuous mapping from a topological space into a discrete topological space is locally constant . ${\displaystyle X}$ ${\displaystyle Y}$

Discrete metrics

A discrete topological space can be determined with a discrete metric ${\displaystyle X}$

${\displaystyle d(x,y):={\begin{cases}0&\mathrm {f{\ddot {u}}r} \ x=y\\1&\mathrm {f{\ddot {u}}r} \ x\neq y\end{cases}}}$

equip that induces the discrete topology.

The equipment with this metric does not offer any significant gain in information. After all, they make terms like Cauchy sequence and completeness applicable.

Proof of the metric axioms

The fulfillment of positive definiteness and symmetry is immediately evident from the definition.

For the verification of the triangle inequality

${\displaystyle d(x,y)\leq d(x,z)+d(z,y)}$

a distinction is made between two cases:

1. If so the left side is equal to 0 and the inequality is definitely fulfilled. ${\displaystyle x=y,}$
2. Is , must, or be, because two different elements cannot match. This means that at least one of the two numbers must be or equal to 1, which is why ${\displaystyle x\neq y}$ ${\displaystyle z\neq x}$ ${\displaystyle z\neq y}$ ${\displaystyle z}$ ${\displaystyle d(x,z)}$ ${\displaystyle d(z,y)}$

   ${\displaystyle d(x,y)=1\leq d(x,z)+d(z,y)}$

applies.

Moreover, it is an ultrametric , because it is only possible with and therefore only with equality . In all other cases so is the tightened triangle inequality ${\displaystyle d}$ ${\displaystyle \max\{d(x,z),d(z,y)\}=0}$ ${\displaystyle d(x,z)=0=d(z,y)}$ ${\displaystyle x=z=y}$ ${\displaystyle 1\leq \max\{d(x,z),d(z,y)\},}$

${\displaystyle d(x,y)\leq \max\{d(x,z),d(z,y)\}}$

applies to all . ${\displaystyle x,y,z\in M}$

Metric properties

With a uniformly discrete metric, a sequence is a Cauchy sequence if and only if it is stationary . Every metric space equipped with a uniformly discrete metric is complete , that is: every Cauchy sequence converges.

Example of a non-uniformly discrete metric

Let the metric space equipped with the amount metric be . Every point has its surroundings ${\displaystyle M:={\bigl \{}{\tfrac {1}{n}}\,{\big |}\,n\in \mathbb {N} {\bigr \}}}$ ${\displaystyle (M,|x-y|).}$ ${\displaystyle {\tfrac {1}{m}}}$

${\displaystyle {\bigl \{}x\in M\;{\big |}\;|x-{\tfrac {1}{m}}|<{\tfrac {1}{m^{2}+m}}{\bigr \}}}$

of all points that are closer to than the point It consists of only one point. Thus, all points are isolated, and the topology induced by is also the discrete. ${\displaystyle {\tfrac {1}{m}}}$ ${\displaystyle {\tfrac {1}{m+1}}.}$ ${\displaystyle {\tfrac {1}{m}}.}$ ${\displaystyle {\tfrac {1}{m}}\in M}$ ${\displaystyle |\cdot |}$

On the other hand, for every one and a point such that, for all the discreetness of the metric which is why no uniform is. ${\displaystyle c\in \mathbb {R} _{>0}}$ ${\displaystyle N:=\lceil {\tfrac {1}{c}}\rceil }$ ${\displaystyle {\tfrac {1}{N}}\in M}$ ${\displaystyle n>N}$   ${\displaystyle |{\tfrac {1}{N}}-{\tfrac {1}{n}}|<{\tfrac {1}{N}}\leq c,}$

It should also be noted that the sequence is a Cauchy sequence that has no limit in . Because it has the subspace topology as a subspace of real numbers , and in has the limit value 0, which does not exist in. ${\displaystyle F:=\left({\tfrac {1}{n}}\right)_{n\in \mathbb {N} }}$ ${\displaystyle M}$ ${\displaystyle M}$ ${\displaystyle \mathbb {R} }$ ${\displaystyle \mathbb {R} }$ ${\displaystyle F}$ ${\displaystyle M}$

Category theoretical background

From the point of view of category theory, the discrete topology on a set is the free topology on this set. To do this, consider the functor from the category of all sets (with all set mappings as morphisms ) to the category of all topological spaces (with all continuous mappings as morphisms), which assigns the discrete topological space to every set and the same mapping between the associated discrete spaces for every set map . This functor is now adjoint left to the forget functor . Usually the images of sets under such functors are called free constructions , for example free groups , free Abelian groups , free modules . In a similar way, the indiscreet topology as a functor is right adjoint to the above mentioned forget functor. That is, the indiscrete topology is the dual term for the discrete topology. ${\displaystyle F\colon \mathbf {Sets} \to \mathbf {Top} }$ ${\displaystyle X}$ ${\displaystyle (X,{\mathcal {P}}(X))}$ ${\displaystyle U\colon \mathbf {Top} \to \mathbf {Sets} }$

Individual evidence

1. ↑ Cigler and Reichel

See also

• Indiscreet topology

literature

• Boto von Querenburg : Set theoretical topology (= Springer textbook ). 3rd, revised and expanded edition. Springer, Berlin et al. 2001, ISBN 3-540-67790-9 .
• Johann Cigler , Hans-Christian Reichel : Topology. A basic lecture (= BI university paperbacks. Vol. 121). Bibliographisches Institut, Mannheim et al. 1978, ISBN 3-411-00121-6 , § 3.1 Metric spaces .