# Math Help - IVT with different topologies

1. ## IVT with different topologies

Prove whether or not the general version of the IVT holds when the range $\mathbb{R}$ is given each of the following topologies:

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

2. Originally Posted by Andreamet
Prove whether or not the general version of the IVT holds when the range $\mathbb{R}$ is given each of the following topologies:

1. The trivial topology
2. The discrete topology
3. The Lower limit topology
Let $f:[a,b] \rightarrow \mathbb{Re}$ 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) ( $\mathbb{R}$ with a discrete and a lower limit topology are totally disconnected).

So, I think IVT holds for 1 only.