Riemann–Siegel formula: Difference between revisions

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In [[topology]], the '''pasting lemma''' is an important result which says that two continuous functions can be "glued together" to create another continuous function. The lemma is implicit in the use of [[piecewise function]]s.
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The pasting lemma is crucial to the construction of the [[fundamental group]] of a topological space; it allows one to concatenate continuous paths to create a new continuous path.
 
== Formal statement ==
 
Let <math>X,Y</math> be both closed (or both open) subsets of a topological space ''A'' such that <math>A = X \cup Y</math>, and let ''B'' also be a topological space. If <math>f: A \to B</math> is continuous when restricted to both ''X'' and ''Y'', then ''f'' is continuous.
 
This result allows one to take two continuous functions defined on closed (or open) subsets of a topological space and create a new one.
 
'''Proof''': if ''U'' is a closed subset of ''B'', then <math>f^{-1}(U )\cap X</math> and <math>f^{-1}(U )\cap Y</math> are both closed since the intersection of two closed sets is closed, and ''f'' restricted to both ''X'' and ''Y'' is continuous. Therefore, their union, <math>f^{-1}(U)</math> is also closed. A similar argument applies when ''X'' and ''Y'' are both open. <math>\Box</math>
 
The infinite analog of this result (where <math>A=X_1\cup X_2\cup X_3\cup\cdots</math>)is not true for closed <math>X_1, X_2, X_3\ldots</math>. It is, however, true if the <math>X_1, X_2, X_3\ldots</math> are open; this follows from the fact that an arbitrary union of open sets is open.
 
== References ==
* [[James Munkres|Munkres, James]]; ''Topology'', Prentice Hall; 2nd edition (December 28, 1999). ISBN 0-13-181629-2.
 
[[Category:Topology]]

Latest revision as of 11:57, 21 November 2014

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