Pasting Lemma

Definition

The pasting lemma is a fundamental result in topology that states that if two continuous functions agree on a shared subset, then there exists a continuous function that combines these functions over their respective domains. This concept is crucial for understanding how spaces can be constructed from simpler pieces and ensures that the properties of continuity are preserved during such constructions.

5 Must Know Facts For Your Next Test

1. The pasting lemma guarantees that the combination of locally defined continuous functions results in a globally defined continuous function when they agree on overlapping regions.
2. It is often used in the construction of new topological spaces by gluing together simpler pieces, like disks or simplices.
3. The lemma highlights the importance of local properties in topology, showing how local continuity can be extended to global continuity.
4. In applications, the pasting lemma can simplify complex problems by breaking them into manageable parts and ensuring their combination maintains continuity.
5. It plays a key role in defining various constructions in algebraic topology, such as the construction of CW complexes or sheaves.

Review Questions

• How does the pasting lemma relate to the concept of continuous functions in topology?
  • The pasting lemma directly connects to continuous functions by establishing that if two continuous functions defined on overlapping domains agree on their shared region, then it is possible to create a new continuous function across their entire combined domain. This illustrates how continuity can be maintained when merging functions, reinforcing the significance of local conditions in determining global properties.
• Discuss how the pasting lemma can be applied to construct new topological spaces from existing ones.
  • The pasting lemma is essential when constructing new topological spaces because it allows mathematicians to glue together simpler spaces while preserving continuity. For instance, when combining open sets or disks in a topological space, ensuring they share common boundaries lets us define a new space without losing the property of being continuous. This method is widely used in algebraic topology and geometric topology.
• Evaluate the implications of the pasting lemma on understanding homeomorphisms and topological equivalence.
  • The pasting lemma's implications extend to homeomorphisms by ensuring that constructions involving continuous functions maintain the structural integrity required for two spaces to be considered topologically equivalent. When constructing complex shapes or surfaces through gluing processes while adhering to the pasting lemma, we can verify that these newly formed spaces retain a homeomorphic relationship with their original components. This understanding strengthens our grasp of how continuity and structure interplay in defining topological equivalence.

Related terms

Continuous Function: A function is continuous if small changes in the input lead to small changes in the output, meaning the pre-image of any open set is open.

Topological Space: A set equipped with a structure that defines which subsets are open, allowing for the study of convergence, continuity, and other topological properties.

Homeomorphism: A continuous function between two topological spaces that has a continuous inverse, indicating that the two spaces are topologically equivalent.