Bonding lemma

The Verklebungslemma ( English Glueing lemma (or Gluing lemma ) or pasting lemma ) is a basic tenet of the mathematical part of the area of general topology . It shows how, under certain conditions, continuous images on topological spaces can be pieced together from those on subspaces and thus, to a certain extent, "glued" together.

Formulation of the lemma

It can be summarized and formulated in general terms as follows:

Let two topological spaces and . ${\displaystyle X}$ ${\displaystyle Y}$

A coverage of and in addition a family of continuous mappings are given . ${\displaystyle (A_{i})_{i\in I}}$ ${\displaystyle X}$ ${\displaystyle f_{i}\colon \,A_{i}\to Y}$

The following may apply:

(1) For and always be . ${\displaystyle i,j\in I}$ ${\displaystyle x\in A_{i}\cap A_{j}}$ ${\displaystyle f_{i}(x)=f_{j}(x)}$

(2) They are either all open subsets or all closed subsets of , whereby in the latter case it should also apply that the family represents a locally finite cover of . ${\displaystyle A_{i}}$ ${\displaystyle X}$ ${\displaystyle (A_{i})_{i\in I}}$ ${\displaystyle X}$

Then:

By the assignment rule

${\displaystyle x\mapsto f(x):=f_{i}(x)\;(i\in I\;,\;x\in A_{i})}$

is an illustration

${\displaystyle f\colon \,X\to Y}$

given and this is continuous.

Inference

The lemma includes the following frequently used criterion:

If a topological space has an open cover or a finite closed cover , then a mapping given on it into another topological space is continuous if and only if each individual restricted mapping is continuous. ${\displaystyle X}$ ${\displaystyle (A_{i})_{i\in I}}$ ${\displaystyle (A_{i})_{i=1,\ldots ,n}\;(n\in \mathbb {N} )}$ ${\displaystyle f\colon \,X\to Y}$ ${\displaystyle Y}$ ${\displaystyle f|_{A_{i}}}$

For proof

The proof of the lemma is essentially based on the following equation which is valid for every subset ${\displaystyle B\subseteq Y}$

${\displaystyle f^{-1}(B)=\bigcup _{i\in I}{f_{i}}^{-1}(B)}$

as well as the fact that (under the respective conditions!) a subset is open (or closed) in if and only if each of the intersections is open (or closed) in . ${\displaystyle A\subseteq X}$ ${\displaystyle X}$ ${\displaystyle A\cap A_{i}\;(i\in I)}$ ${\displaystyle A_{i}}$

See also

• Final topology

literature

• 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 ).
• Fred H. Croom: Principles of Topology . Saunders, Philadelphia, 1989, ISBN 0-03-012813-7 .
• Lutz Führer : General topology with applications . Vieweg Verlag, Braunschweig 1977, ISBN 3-528-03059-3 .
• IM Singer , JA Thorpe : Lecture Notes on Elementary Topology and Geometry (=  Undergraduate texts in mathematics ). Springer Verlag, New York / Heidelberg / Berlin 1976, ISBN 0-387-90202-3 ( MR0413152 ).
• Horst Schubert : Topology . 4th edition. BG Teubner Verlag, Stuttgart 1975, ISBN 3-519-12200-6 ( MR0423277 ).

References and footnotes

1. ↑ In the textbook by Camps / Kühling / Rosenberger (pp. 57 & 519) there is also talk of an [em] continuation sentence in connection with this theorem .
2. ^ ^{A b} Fred H. Croom: Principles of Topology. 1989, p. 151
3. ^ ^{A b} I. M. Singer, JA Thorpe: Lecture Notes on Elementary Topology and Geometry. 1976, p. 51
4. ^ Lutz Führer: General topology with applications. 1977, p. 43
5. ↑ Thorsten Camps et al .: Introduction to set theoretical and algebraic topology. 2006, p. 57
6. ↑ In the specialist literature - for example at Camps / Kühling / Rosenberger as well as Croom and Singer / Thorpe - the case of coverage with two subsets is often considered alone.
7. ↑ This always means continuity in relation to the induced subspace topology .
8. ↑ Horst Schubert: Topology. 1975, pp. 27-28