borkborkmath
New member
- Joined
- Mar 4, 2011
- Messages
- 16
Let \(\displaystyle \alpha\) be an element in the index set I
Let X = \(\displaystyle \prod\)_{\(\displaystyle \alpha\)}X_\(\displaystyle \alpha\) be the topological product of the family of spaces {X_\(\displaystyle \alpha\)}. Prove that a functon f:Y->X from a space Y into the product X is continuous if and only if for each \(\displaystyle \alpha\) \(\displaystyle \in\) I the function f_\(\displaystyle \alpha\) = p_\(\displaystyle \alpha\)f: Y->X_\(\displaystyle \alpha\)
I think p_i is defined in a product space as the ith projection p_i: X -> X_i such that p_i(a) = a_i.
Wouldn't this be true because of the definition of continuity? Since a function is uniformly continuous if and only if it is continuous at all points?
Let X = \(\displaystyle \prod\)_{\(\displaystyle \alpha\)}X_\(\displaystyle \alpha\) be the topological product of the family of spaces {X_\(\displaystyle \alpha\)}. Prove that a functon f:Y->X from a space Y into the product X is continuous if and only if for each \(\displaystyle \alpha\) \(\displaystyle \in\) I the function f_\(\displaystyle \alpha\) = p_\(\displaystyle \alpha\)f: Y->X_\(\displaystyle \alpha\)
I think p_i is defined in a product space as the ith projection p_i: X -> X_i such that p_i(a) = a_i.
Wouldn't this be true because of the definition of continuity? Since a function is uniformly continuous if and only if it is continuous at all points?