Chainset (General Topology)

In general topology , one of the branches of mathematics , the chain theorem deals with the question under which conditions in a topological space the union of connected subspaces is itself connected.

Formulation of the sentence

The sentence can be formulated as follows:

A topological space and a family of connected subspaces are given. ${\displaystyle X}$ ${\displaystyle (A_{i})_{i\in I}}$

Let the subspace family be concatenated in the following sense:

For every two indices there is always a finite subfamily with: ${\displaystyle p,q\in I}$ ${\displaystyle (A_{i_{1}},\ldots ,A_{i_{n}})\;(n\in \mathbb {N} )}$

(a) and ${\displaystyle A_{p}=A_{i_{1}}}$ ${\displaystyle A_{q}=A_{i_{n}}}$

(b) Any two consecutive sets of the finite subfamily may overlap ; For always apply . ${\displaystyle r=1,\ldots ,n-1}$ ${\displaystyle A_{i_{r}}\cap A_{i_{r+1}}\neq \emptyset }$

Then:

The Union

${\displaystyle V=\bigcup _{i\in I}A_{i}}$

forms a contiguous subspace of . ${\displaystyle X}$

Tightening

The above condition (b) can be - with the same assertion! - weaken to the effect that one only demands the following:

(b ') Of every two consecutive subspaces of the finite subfamily , at least one of the two always contains a point of contact of the other; ma W .: for always or . ${\displaystyle r=1,\ldots ,n-1}$ ${\displaystyle {\overline {A_{i_{r}}}}\cap A_{i_{r+1}}\neq \emptyset }$ ${\displaystyle A_{i_{r}}\cap {\overline {A_{i_{r+1}}}}\neq \emptyset }$

Inferences

The chain set pulls - even in its simple version! - the following results immediately after:

(1) If in a topological space a family of connected subspaces has non-empty intersections , the union of these subspaces in turn forms a connected subspace.

(2) If two points of a topological space are contained in a connected subspace of this space, then this space is connected.

(3) In a topological space, the connected component of a point is equal to the union of all those connected subspaces which contain this point, i.e. the largest of all connected subspaces to which this point belongs .

In the tightened version of the chain sentence, the following result is immediately obtained:

(4) In a topological space, a union of connected subspaces, in which at least one of the two always contains a contact point of the other, forms a connected subspace.

literature

• P. Alexandroff , H. Hopf : Topology . First volume. Corrected reprint (=  The basic teachings of the mathematical sciences in individual representations with special consideration of the areas of application . Volume 45 ). Springer Verlag, Berlin / Heidelberg / New York 1974 ( MR0185557 ).
• Thorsten Camps , Stefan Kühling , Gerhard Rosenberger : Introduction to set-theoretical and algebraic topology (=  Berlin study series on mathematics . Volume 15 ). Heldermann Verlag, Lemgo 2006, ISBN 3-88538-115-X ( MR2172813 ).
• Lutz Führer : General topology with applications . Vieweg Verlag, Braunschweig 1977, ISBN 3-528-03059-3 .
• KD Joshi : Introduction to General Topology . Wiley Eastern Limited, New Delhi / Bangalore / Bombay / Calcutta 1983, ISBN 0-85226-444-5 .
• Willi Rinow : Textbook of Topology (=  university books for mathematics . Volume 79 ). VEB Deutscher Verlag der Wissenschaften, Berlin 1975 ( MR0514884 ).
• Horst Schubert : Topology . 4th edition. BG Teubner Verlag, Stuttgart 1975, ISBN 3-519-12200-6 ( MR0423277 ).

References and comments

1. ↑ ^{a b} Thorsten Camps et al .: Introduction to set-theoretical and algebraic topology. 2006, p. 87
2. ↑ ^{a b c} Lutz Führer: General topology with applications. 1977, p. 86
3. ↑ P. Alexandroff, H. Hopf: Topology. 1974, p. 48
4. ↑ ^{a b} Willi Rinow: Textbook of Topology. 1975, pp. 141-142
5. ↑ Horst Schubert: Topology. 1975, p. 38
6. ↑ In fact, the chain proposition in its simple version also follows directly from (1); see. Thorsten Camps et al., Op. Cit., Pp. 86-87.
7. ↑ ^{a b} Alexandroff / Hopf, op.cit., P. 49
8. ↑ Thorsten Camps et al., Op.cit., P. 94
9. ↑ Schubert, op.cit., P. 39
10. ^ KD Joshi: Introduction to General Topology. 1983, p. 145