Intuitively, what is the difference between a "simply connected" set and a "locally connected" set?

Question

I was looking into the Mandelbrot Set and saw a note that said it has been proven that the Mandelbrot Set is "simply connected" but it is still an open question of whether or not it is "locally connected" (MLC).

I can easily understand what simply connected means intuitively--you can draw a line between any two points in the set, it has no holes, etc.

But I'm having trouble understanding visually what "locally connected" means, and what the difference is between simple connection and local connection. Every page I see on local connectedness only describes it in topological jargon that goes over my head.

Is there any way to intuitively or visually describe what the difference is between a locally and simply connected set?

Answer 1

Locally connected intuitively means that when you "make a sufficient amount of zoom" in any part of the object, youre gonna always "see" one total piece rather than a lot of small pieces. For example the famous "topologist sine curve" is one space that is NOT locally connected; no matter how much you zoom in the $Y$-axis you always gonna see a lot of separated bars, rather than "one" object.

Answer 2

A locally connected space is a space where every neighborhood of every point $x$ contains an connected open subset containing $x$. Informally, something preventing a space from being locally connected would be a neighborhood of a point with "disconnected parts arbitrarily close to that point". Here is an attempt at a drawing of a connected space which is not locally connected (the dashed red line is a neighborhood of the red mark showing that the space is not locally connected). It is the comb space.