Thread: IVT with different topologies

Prove whether or not the general version of the IVT holds when the range is given each of the following topologies:

1. The trivial topology
2. The discrete topology
3. The Lower limit topology

Originally Posted by Andreamet

Prove whether or not the general version of the IVT holds when the range is given each of the following topologies:

1. The trivial topology
2. The discrete topology
3. The Lower limit topology

Let be a continuous function that IVT holds.

Since the image of a (path) connected space under a continuous function is (path) connected, IVT does not hold for (2) and (3) ( with a discrete and a lower limit topology are totally disconnected).

So, I think IVT holds for 1 only.