I don't think I understand what 'locally connected' means. You can easily choose a rectanglular area that contains 2 areas of the set that are not joined.
Strictly speaking, you would do this coloring with all connected components of the intersection of your rectangle and M. (And the rectangle could be any region.)
The example messes this up, although it is a correct example, the square on the diagram showing the local piece containing non-connected parts is wrong. The comb has more and more teeth, infinitely many in a bounded space, on the left side. Only a rectangle which includes the left edge properly shows why the set isn't locally connected. The rectangle pictured includes finitely many teeth which have a separation between them. A rectangle overlapping the left edge of the comb would include separate components which get arbitrarily close to that left edge and so can't be separated by a border.
-take any rectanglular section of the complex plain that includes part or all of the Mandelbrot set
-draw the Mandelbrot set in black
-pick an arbitary black point and colour it red
-recursively colour every black point touching a red point (flood fill)
Then every black point would be recoloured red. And this would work with a pixel based image of the mandelbrot if the image had a high enough resolution. Is that right?
Do those first four steps. You wouldn't (necessarily) cover every black point in your rectangle. Choose a remaining black point and flood fill from that, say green. Keep on doing this with different colours until you've covered every black point in your rectangle. You have a bunch of regions of different colours.
Now, if the different coloured regions are all nicely separate, then your set is locally connected. Because each point is either cleanly in one component or cleanly in the other.
If on the other hand your drawing looks like https://commons.m.wikimedia.org/wiki/File:Julia_set_for_the_... with mixed up boundaries where some points are infinitesimally close to more than one colour, then it's not locally connected.
The difficulty with intuition is that in our intuition, coloured regions always have reasonable boundaries (think countries in a map: the border can be wiggly but there's never infinitely many tiny bits of one country mixed up in the boundary of two others). In fractal geometry, things like the Newton fractal picture above are quite usual.
I still don't understand.
https://en.wikipedia.org/wiki/Locally_connected_space
Unfortunately I don't have the slightest idea what it actually means... that article does not have any ELI5 sentence within it.
Not just me then? ;0)
"MLC posits that the Mandelbrot set isn’t just connected; it’s locally connected — no matter how much you zoom in on the Mandelbrot set, it will always look like one connected piece. For instance, a circle is locally connected. An extremely fine-toothed comb, on the other hand, is not. Though the entire shape is connected, if you skip over the shaft and instead zoom in on the tips of some of its teeth, you’ll just see a bunch of separate line segments."