In which of the three topologies is X locally connected?

Question

Question: In which of the three topologies is X locally connected ?

Below is a link which shows that all the topologies are connected...

In which of the three topologies is X connected?

Below is how I showed that all of them are locally connected.

It would be great if someone helps me on this. Appreciate your help & support.

Answer 1

1. Not locally connected at $(0,0)$, using an argument similar to the topologist’s sine curve.

2. Locally connected because locally an interval except at the top, where the same argument that gives connectedness also gives local connectedness.

3. Is a quotient image of a locally connected space, so locally connected.