Now pick a point somewhere on the left-most vertical spike (but not at the point where that spike meets the horizontal interval) and look at any disc small enough to not contain any piece of the horizontal interval. Since there are infinitely many points of the form arbitrarily close to on the horizontal interval, the disc will contain a piece of each of these infinitely many separated vertical spikes. That’s true no matter how small is, so the comb space is not locally connected at any point on the left-most vertical spike.