Is there a continuous function from Ropfu onto RopfL From Ro

Is there a continuous function from Ropf_u onto Ropf_L? From Ropf_FC onto Ropf_L?

Solution

4. We need to show that,,,if Y is a subspace of R with more than one point, then Y is not connected. To do this, let Y be such a subspace, and let x,yY with x<y Let L={zY:z<y} and R={zY:zy}. It follwos that Y=LR , LR=, and L and R are both open in the subspace Y. continuous image of a connectd set is connected but Y is not connected, because {L,R} forms seperation of Y. Hence only continuous function from Ru onto Rl is continous is constant function.