Filter convergence

The filter convergence is a convergence term in topology , a branch of mathematics . It is formalized using set filters and, in addition to the convergence of networks, is a possibility to generalize the convergence of sequences in topological spaces .

The need to generalize the convergence of sequences results from the fact that the use of sequences in topological spaces is not sufficient to characterize topological properties. Thus, for example, functions construct which the topological characterization of continuity (archetypes of open sets are open again) are not enough, but for which the classical characterization in metric spaces is valid (the sequence converges against so converges against ). The filter convergence generalizes the sequence convergence, so that topological properties can also be characterized in topological spaces via convergence and the terms derived from it. ${\displaystyle (x_{n})_{n\in \mathbb {N} }}$ ${\displaystyle x}$ ${\displaystyle f(x_{n})}$ ${\displaystyle f(x)}$

Framework conditions and problems

If a metric space is given, then a sequence is called convergent if: ${\displaystyle (X,d)}$ ${\displaystyle (x_{n})_{n\in \mathbb {N} }}$

for each exists one , so that for all . ${\displaystyle \varepsilon >0}$ ${\displaystyle N\in \mathbb {N} }$ ${\displaystyle d(x_{n},x)\leq \varepsilon }$ ${\displaystyle n\geq N}$

This formalizes the intuitive notion that a convergent sequence comes as close as desired to its limit value: for any given distance , at some point all elements of the sequence are closer to the limit value than this distance. ${\displaystyle \varepsilon }$

Every metric space is always a topological space . The open sets of the topology are then exactly the unions of (any number of) open spheres with a variable radius . Topological terms such as closure , continuity and compactness are thus well defined in metric spaces and can be described in two equivalent ways. The first is called topological characterization in this article , and the other is characterization by sequences . If one looks at the seclusion as an example, the following applies: ${\displaystyle (X,{\mathcal {O}}_{d})}$ ${\displaystyle {\mathcal {O}}_{d}}$ ${\displaystyle B_{r}(x)=\{y\in X\mid d(y,x)<r\}}$ ${\displaystyle r}$

• topological characterization : is the complement of lies in . ${\displaystyle A}$ ${\displaystyle \iff }$ ${\displaystyle A}$ ${\displaystyle {\mathcal {O}}_{d}}$
• Characterization by sequences : is complete The limit of each convergent sequence from is again in . ${\displaystyle A}$ ${\displaystyle \iff }$ ${\displaystyle A}$ ${\displaystyle A}$

The definition of the convergence of sequences can easily be transferred to any topological space. For this purpose, the distance from the limit value is understood as the environment of the limit value and then extended to any environment of the limit value in the context of the transmission . A sequence in a topological space is called convergent to if: ${\displaystyle \varepsilon }$ ${\displaystyle \varepsilon }$ ${\displaystyle (x_{n})_{n\in \mathbb {N} }}$ ${\displaystyle x}$

for every environment of exists one , so that for all . ${\displaystyle U}$ ${\displaystyle x}$ ${\displaystyle N\in \mathbb {N} }$ ${\displaystyle x_{n}\in U}$ ${\displaystyle n\geq N}$

In topological spaces, the topological characterization and the characterization by sequences of topological properties generally do not match. There are cases of points that are in the closure of a set but are not reached by any sequence in the set, as well as points of contact against which no sequence converges. For this reason, a distinction is made between the two types of characterization in topological spaces. The characterization by sequences is given the prefix "follow-" (sequence closed, sequence compact, etc.), while the names of the topological characterization mostly remain unchanged (with the exception of the coverage compactness ).

Thus, on the one hand, sequences are only suitable to a limited extent for the investigation of topological structures, on the other hand, they are also a popular and intuitively accessible tool for many proofs. The filter convergence now generalizes the concept of sequence convergence, so that the equivalence described above of characterization by sequences (and later filters) and topological characterization, as in metric spaces, also applies in any topological spaces. The sequence convergence thus proves to be a special case of the filter convergence.

definition

A topological space is given . Let be a set filter in and be the environment filter of , i.e. the set of all environments of ${\displaystyle (X,{\mathcal {O}})}$ ${\displaystyle {\mathcal {F}}}$ ${\displaystyle X}$ ${\displaystyle {\mathcal {U}}(x)}$ ${\displaystyle x\in X}$ ${\displaystyle x}$

The filter is called convergent to if is. You then write and name a Limes point of . ${\displaystyle {\mathcal {F}}}$ ${\displaystyle x}$ ${\displaystyle {\mathcal {F}}\supset {\mathcal {U}}(x)}$ ${\displaystyle {\mathcal {F}}\rightarrow x}$ ${\displaystyle x}$ ${\displaystyle {\mathcal {F}}}$

Applies to everyone and everyone that is, that's what a point of contact is called. Thus the set of all points of contact is given as ${\displaystyle U\in {\mathcal {U}}(x)}$ ${\displaystyle F\in {\mathcal {F}}}$ ${\displaystyle F\cap U\neq \emptyset }$ ${\displaystyle x}$ ${\displaystyle B}$

${\displaystyle B=\bigcap _{F\in {\mathcal {F}}}{\overline {F}}}$.

Here denotes the conclusion of the set . ${\displaystyle {\overline {F}}}$ ${\displaystyle F}$

Example: transition to sequence convergence

Sequence convergence is a special case of filter convergence. If a sequence is given, then one defines ${\displaystyle (x_{n})_{n\in \mathbb {N} }}$

${\displaystyle R_{N}:=(x_{n})_{n\geq N}}$,

the sequence without the first sequence members. If you choose all of these as the filter base , you get the filter belonging to the sequence ${\displaystyle N}$ ${\displaystyle R_{N}}$

${\displaystyle {\mathcal {F}}=\{F\subset X\mid {\text{Es gibt ein }}N\in \mathbb {N} ,{\text{ so dass }}R_{N}\subset F{\text{ ist }}\}}$.

The convergence of the sequence against is now equivalent to after the section "Framework conditions and problems" ${\displaystyle x}$

for every environment of exists one such that , ${\displaystyle U}$ ${\displaystyle x}$ ${\displaystyle N\in \mathbb {N} }$ ${\displaystyle R_{N}\subset U}$

since, by definition, contains all members of the sequence with an index greater than . But it follows directly from this that , there is. So then . ${\displaystyle R_{N}}$ ${\displaystyle N}$ ${\displaystyle U\in {\mathcal {F}}}$ ${\displaystyle R_{N}\subset U}$ ${\displaystyle {\mathcal {U}}(x)\subset {\mathcal {F}}}$

If a sequence converges against , the filter belonging to the sequence also converges against . The limit point of the filter and the limit value of the sequence then match. Analogously, one shows that the contact points of the filter are exactly the accumulation points of the sequence. ${\displaystyle x}$ ${\displaystyle x}$

Inferences

The following statements can then be made directly about the filter convergence:

• A is contained in the closure of the set if and only if there is a filter that contains the set and converges to it. ${\displaystyle x}$ ${\displaystyle A}$ ${\displaystyle A}$ ${\displaystyle x}$
• A picture is exactly then steadily in when for each filter , the opposite converges the image filter to converge. The image filter is defined as the filter in the image space that has the filter base . ${\displaystyle f}$ ${\displaystyle X}$ ${\displaystyle {\mathcal {F}}}$ ${\displaystyle x}$ ${\displaystyle f({\mathcal {F}})}$ ${\displaystyle f(x)}$ ${\displaystyle \{f(F)\mid F\in {\mathcal {F}}\}}$

The statements about sequence convergence as they apply in metric spaces are therefore transferred almost literally to filter convergence and then also apply in topological spaces.

With the filter convergence , further properties can be characterized: A topological space is a Hausdorff space if and only if every convergent filter has exactly one Limes point or a topological space is compact if and only if every ultrafilter converges.

literature

• Boto von Querenburg : Set theoretical topology . 3. Edition. Springer-Verlag, Berlin Heidelberg New York 2001, ISBN 978-3-540-67790-1 , doi : 10.1007 / 978-3-642-56860-2 .
• Dirk Werner : Functional Analysis . 7th, corrected and enlarged edition. Springer-Verlag, Heidelberg Dordrecht London New York 2011, ISBN 978-3-642-21016-7 , doi : 10.1007 / 978-3-642-21017-4 .

Individual evidence

1. ↑ von Querenburg: Set theoretical topology. 2011, p. 74.
2. ^ Werner: Functional Analysis. 2011, p. 405.
3. ↑ von Querenburg: Set theoretical topology. 2011, p. 74.