Locally connected continuum

A continuum that is a locally connected space. Examples of locally connected continua are the -dimensional cube, the Hilbert cube, and all Tikhonov cubes (cf. Tikhonov cube). The union of the graph of the function

and the interval gives an example of a continuum that is not locally connected (at the points of ). A metrizable continuum is locally connected if and only if it is a curve in the sense of Jordan (cf. Line (curve)). Any metrizable locally connected continuum is path-connected (cf. Path-connected space). Moreover, any two distinct points of such a continuum are contained in a simple arc lying in .