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The MLC Conjecture - Mandelbrot Locally Connected - suggests that the fractal, known as the Mandelbrot Set is 'locally connected'.
'Locally Connected' in this instance is defined thus :
A topological space is locally connected at the pointif every neighborhood of
contains a connected open neighborhood. It is called locally connected if it is locally connected at every point.“
Source Wolfram Mathworld
The conjecture remains 'open' - i.e has neither been proved or disproved.
See Wikipedia
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