Network (topology)

A network or a Moore-Smith sequence is a generalization of a sequence in topology (a branch of mathematics ) . The term goes back to Eliakim H. Moore and Herman L. Smith , who introduced it in 1922. With so-called Cauchy nets, the concept of completeness of metric spaces can be generalized to uniform spaces . In addition, they can be used in integral calculus to describe the Riemann integrability .

motivation

It should be explained briefly beforehand why a generalization of consequences is necessary. In a metric space , the topology can be completely by sequence convergence characterize: A subset is exactly then finished if for every sequence in with the following applies: . Properties such as continuity of functions and compactness can also be defined using sequences (e.g. coverage compactness and sequence compactness are equivalent in metric spaces ). ${\ displaystyle (X, d)}$${\ displaystyle A \ subseteq X}$${\ displaystyle (x_ {n})}$${\ displaystyle A}$${\ displaystyle \ lim \ textstyle _ {n \ rightarrow \ infty} {x_ {n}} = x}$${\ displaystyle x \ in A}$

In topological spaces, however, a subset is no longer necessarily closed if each sequence has a limit value in (e.g. in is not closed with the order topology , although the limit value in is also in for every convergent sequence in ). ${\ displaystyle A}$${\ displaystyle A}$${\ displaystyle A = [0, \ omega _ {1} [}$${\ displaystyle X = [0, \ omega _ {1}]}$${\ displaystyle A}$${\ displaystyle A}$

Here networks represent a more meaningful generalization: A subset of a topological space is closed if and only if every network in that converges in has a limit value in . As in metric spaces, continuity can also be defined by replacing “sequence” with “network” (see below; note that there is no equivalent definition in terms of sequences for continuity in topological spaces). ${\ displaystyle A \ subseteq X}$${\ displaystyle (X, {\ mathfrak {T}} _ {X})}$${\ displaystyle A}$${\ displaystyle X}$${\ displaystyle A}$

Also, a set is compact if and only if every network has a convergent subnet.

Definitions

For a directed set and a set , a network is a picture . Usually you write analogously to episodes . Since the natural numbers form a directed set with the usual arrangement, sequences are special networks. ${\ displaystyle (I, \ triangleleft _ {I})}$${\ displaystyle X}$${\ displaystyle x \ colon I \ to X}$${\ displaystyle (x_ {i}) _ {i \ in I}}$

Subnet

${\ displaystyle (I, \ triangleleft _ {I})}$and let directed sets, a network in, and a map which satisfies the following condition: ${\ displaystyle (J, \ triangleleft _ {J})}$${\ displaystyle (x_ {i}) _ {i \ in I}}$${\ displaystyle X}$${\ displaystyle \ varphi \ colon J \ to I}$

${\ displaystyle \ forall i_ {0} \ in I \ \ exists j_ {0} \ in J \ \ forall j \ triangleright _ {J} j_ {0} \ colon \ \ varphi (j) \ triangleright _ {I} i_ {0}}$

(Such a mapping is called confinal ). Then the network is called a subnet of the network . ${\ displaystyle \ varphi}$${\ displaystyle (x _ {\ varphi (j)}) _ {j \ in J}}$${\ displaystyle (x_ {i}) _ {i \ in I}}$

Convergent network

If a topological space is defined as for sequences: A network is said to be convergent to if: ${\ displaystyle X}$${\ displaystyle x: = (x_ {i}) _ {i \ in I}}$${\ displaystyle z \ in X}$

${\ displaystyle \ forall U \ in {\ mathcal {U}} (z) \ \ exists i_ {0} \ in I \ \ forall i \ in I: i_ {0} \ triangleleft i \ Rightarrow x_ {i} \ in U}$,

where denotes the environment filter of . One then writes or or The formal definition can be paraphrased as follows: For every neighborhood of there is an initial index in the directed set , so that members of the network with index after are contained in the presented environment. ${\ displaystyle {\ mathcal {U}} (z)}$${\ displaystyle z}$${\ displaystyle (x_ {i}) _ {i \ in I} \ to z}$${\ displaystyle x_ {i} \ to z}$${\ displaystyle \ textstyle z = \ lim _ {i \ in I} x_ {i}.}$${\ displaystyle z}$${\ displaystyle i_ {0}}$${\ displaystyle I}$${\ displaystyle i}$${\ displaystyle i_ {0} \; (i \ triangleright i_ {0})}$

The concept of convergence can be traced back to the convergence of a filter : For this purpose, the section filter is defined as that of the filter base

${\ displaystyle \ left \ {\ left \ {x_ {j} \ mid j \ triangleright i \ right \} \ mid i \ in I \ right \}}$

generated filters. The network converges to a point if and only if the associated section filter converges to, i.e. H. contains the environment filter of. ${\ displaystyle z \ in X}$${\ displaystyle z}$${\ displaystyle z}$

Cluster point

A point is called an accumulation point of a network if and only if: ${\ displaystyle z \ in X}$${\ displaystyle x,}$

${\ displaystyle \ forall U \ in {\ mathcal {U}} (z) \ \ forall i \ in I \ \ exists j \ triangleright i \ x_ {j} \ in U}$,

d. H. every area around is reached at any position in the filter. Again, a characterization via the section filter is possible: is the accumulation point of a network if and only if it is the point of contact of the section filter, i. H. if the intersection of every environment with every element of the filter is not empty. ${\ displaystyle z}$${\ displaystyle z}$

A further characterization is possible via subnetworks: is the accumulation point of a network if and only if there is a subnetwork that converges to. ${\ displaystyle z}$${\ displaystyle z}$

Cauchynetz

If a uniform space is defined: A net on is called Cauchy net if there is an index for every neighborhood , so that all pairs of members of the net with later indices are of the order adjacent, i.e. that is , that applies. In formulas: ${\ displaystyle (X, \ Phi)}$${\ displaystyle x: = (x_ {i}) _ {i \ in I}}$${\ displaystyle X}$${\ displaystyle N \ in \ Phi}$${\ displaystyle i_ {0} \ in I}$${\ displaystyle j, k \ triangleright i_ {0}}$${\ displaystyle N}$${\ displaystyle (x_ {j}, x_ {k}) \ in N}$

${\ displaystyle \ forall N \ in \ Phi \; \ exists i_ {0} \ in I \; \ forall j, k \ triangleright i_ {0} \ colon \; (x_ {j}, x_ {k}) \ in N.}$

Two Cauchy nets and are considered equivalent , in characters if ${\ displaystyle x: = (x_ {i}) _ {i \ in I}}$${\ displaystyle y: = (y_ {i}) _ {i \ in I}}$${\ displaystyle x \ sim y}$

${\ displaystyle \ forall N \ in \ Phi \; \ exists i_ {0} \ in I \; \ forall j, k \ triangleright i_ {0} \ colon \; (x_ {j}, y_ {k}) \ in N.}$

The completion of is ${\ displaystyle (X, \ Phi)}$

${\ displaystyle C / \! \ sim}$

with than the set of all Cauchy nets. All Cauchy nets converge in a complete space and equivalent Cauchy nets have the same limit. ${\ displaystyle C}$

completeness

A uniform space is complete if and only if every Cauchy network is convergent on . ${\ displaystyle X}$${\ displaystyle X}$

An example of a complete uniform space is the pro-finite numbers a completion of the uniform space of the whole numbers${\ displaystyle {\ widehat {\ mathbb {Z}}},}$${\ displaystyle \ mathbb {Z}.}$

Applications

Definition of the closed envelope

If a subset of the topological space is then a point of contact of (ie contained in the closed envelope of ) if and only if there is a network with terms that converges to. ${\ displaystyle A}$${\ displaystyle X}$${\ displaystyle y \ in X}$${\ displaystyle A}$${\ displaystyle A}$${\ displaystyle (x_ {i}) _ {i \ in I}}$${\ displaystyle x_ {i} \ in A}$${\ displaystyle y}$

Local definition of continuity
• Be and topological spaces. A mapping is continuous at the point if and only if the following applies to every network in : From follows .${\ displaystyle X}$${\ displaystyle Y}$${\ displaystyle f \ colon X \ to Y}$${\ displaystyle x \ in X}$${\ displaystyle (x_ {i}) _ {i \ in I}}$${\ displaystyle X}$${\ displaystyle x_ {i} \ to x}$${\ displaystyle f (x_ {i}) \ to f (x)}$
Riemann integral

The amount of the decompositions of the real interval , is determined by the inclusion to a targeted amount:  : contains all the points of . For a real-valued bounded function on be by the upper sum ${\ displaystyle {\ mathcal {Z}}}$${\ displaystyle Z: = (x_ {0}, x_ {1}, x_ {2}, \ dotsc, x_ {n})}$ ${\ displaystyle [a, b]}$${\ displaystyle a = x_ {0} ${\ displaystyle Z_ {1} \ triangleleft Z_ {2}}$${\ displaystyle Z_ {2}}$${\ displaystyle Z_ {1}}$${\ displaystyle [a, b]}$

${\ displaystyle \ mathbf {O} (f) \ colon {\ mathcal {Z}} \ to \ mathbb {R}; (x_ {0}, x_ {1}, x_ {2}, \ dotsc, x_ {n }) \ mapsto \ sum _ {j = 1} ^ {n} (x_ {j} -x_ {j-1}) \ cdot \ sup _ {x \ in [x_ {j-1}, x_ {j} ]} f (x)}$

and the sub-total

${\ displaystyle \ mathbf {U} (f) \ colon {\ mathcal {Z}} \ to \ mathbb {R}; (x_ {0}, x_ {1}, x_ {2}, \ dotsc, x_ {n }) \ mapsto \ sum _ {j = 1} ^ {n} (x_ {j} -x_ {j-1}) \ cdot \ inf _ {x \ in [x_ {j-1}, x_ {j} ]} f (x)}$

two networks defined. The function is exactly then Riemann-integrable on when both networks against the same real number converge. In the case it is . ${\ displaystyle f}$${\ displaystyle [a, b]}$${\ displaystyle c}$${\ displaystyle c = \ int _ {a} ^ {b} f (x) \, \ mathrm {d} x}$

Instead of the upper and lower sums, Riemann sums can also be used to characterize the Riemann integrability. A more complicated directed set is needed for this. An element of this set always consists of a decomposition as above and an intermediate vector of intermediate points belonging to the decomposition . The order on is now defined in such a way that an element is really smaller than if it is a real subset of . ${\ displaystyle {\ mathcal {I}}: = \ left \ {\ left ((x_ {0}, x_ {1}, \ dotsc, x_ {n}), (t_ {1}, \ dotsc, t_ { n}) \ right): t_ {j} \ in [x_ {j-1}, x_ {j}] \ right \}}$${\ displaystyle (t_ {1}, \ dotsc, t_ {n})}$${\ displaystyle {\ mathcal {I}}}$${\ displaystyle (Z, t)}$${\ displaystyle (Z ', t')}$${\ displaystyle Z}$${\ displaystyle Z '}$

A function can be Riemann-integrated if and only if the network ${\ displaystyle f}$

${\ displaystyle {\ mathcal {I}} \ to \ mathbb {R} \ colon \ left ((x_ {0}, \ dotsc, x_ {n}), (t_ {1}, \ dotsc, t_ {n} ) \ right) \ mapsto \ sum _ {j = 1} ^ {n} (x_ {j} -x_ {j-1}) f (t_ {j})}$

converges. The limit value is then the Riemann integral.

This approach is more complicated than that with upper and lower sums, but it also works with vector-valued functions.

Individual evidence

1. ^ EH Moore, HL Smith: A General Theory of Limits . In: American Journal of Mathematics . 44, No. 2, 1922, pp. 102-121. doi : 10.2307 / 2370388 .