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 point if 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