[nonsensikal]

You don't get to choose a rectangle, you choose a point.

[hermitcrab]

The doesn't seem to fit with the comb analogy.

I still don't understand.

[nhatcher]

Non locally connected spaces are a bit pathological. Means that given a point there is always a neighborhood of the point (might be very small) that is connected. An example of a connected but not locally connected is: https://en.m.wikipedia.org/wiki/Topologist%27s_sine_curve From (0, O) any neighborhood, no matter how small contains points that belong to the curve but cannot reach (0, 0) and stay in the neighborhood.

[returningfory2]

I think this part of the article is incorrect.

[matt-noonan]

It's almost correct, but misses the point in an annoying way that kind of ruins the example. What does work is something like the subset of the plane given by { (x, y) | x real, y rational } U { (0, y) | y real }. This is connected, because you can walk from any point (x,y) to any other point (x',y') by traveling horizontally to the Y axis at (0,y), vertically to (0,y'), then horizontally to (x',y'). But it isn't locally connected away from the Y axis because for a tiny enough open set S around a point (x,y), there are other points in S that you can't get to from (x,y) without leaving S.