See below the 2 sets A and B and show the following:
1- Show that A and B are each connected but A and B are not separated.
2- Show that A union B is connected set.
3- Show that A union B is not locally connected at p (0,1)
4- Provide an example of a topological space X which is locally connected at p(0,1) and another example which is locally connected at each point of X.
Please use the following:
A={(0,y) : 1/2<=y<=1}
B={(x,y): y=sin(1/x), 0