I Can we have a pasting lemma for uniform continuous functions

PKSharma
Messages
2
Reaction score
0
In analysis, the pasting or gluing 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 functions. Can we have a similar situation for uniform continuous functions?
 
Physics news on Phys.org
The result hold when X has the property that every ball is path connected.
 
PKSharma said:
In analysis, the pasting or gluing 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 functions. Can we have a similar situation for uniform continuous functions?

In what context are you working? Real numbers? Connected metric/topological spaces?

For example, gluing two closed intervals keeps things uniform continuous. I guess the finite union of compacts will work too.
 
A sphere as topological manifold can be defined by gluing together the boundary of two disk. Basically one starts assigning each disk the subspace topology from ##\mathbb R^2## and then taking the quotient topology obtained by gluing their boundaries. Starting from the above definition of 2-sphere as topological manifold, shows that it is homeomorphic to the "embedded" sphere understood as subset of ##\mathbb R^3## in the subspace topology.
Back
Top