What's the definition of a partition? I'm going to assume it's not just "sets whose union is the space", else a counterexample is and
Let s.t and .Suppose that is a partition of a topological space and that is a map to another space . Prove that if the restrictions and are both continuous then is continuous.
I need to show that and are connected (since if is continuous on and , then the only place it might be discontinuous is at the boundary, where meets ).
is connected (as is ) since is a continuous map between any two points.
This is what I know, but I can't see what I can do from here! Can someone help?
I know that A and B are open. Since f is continuous then are also open.
. This is the union of two open sets so is open.
Therefore for any as defined before, is also going to be open so maps an open set to an open set. Therefore f is continuous.
I think that's about right. The odd thing is that this is in the connectedness chapter, that's why I was trying to approach it from a connectedness angle.
Sorry I keep getting this wrong! I think i'm getting either flustered or tired.
let be open.
If or then is open since we know that is continuous on these intervals.
If has points in both and then partition into two sets where and .
are both open since is continuous on both and .
Hence is always open. The preimage of an open set is also an open set so is continuous.