Locally connected continuum

A continuum that is a locally connected space. Examples of locally connected continua are the $ n $- dimensional cube, $ n = 0 , 1 \dots $ the Hilbert cube, and all Tikhonov cubes (cf. Tikhonov cube). The union of the graph of the function

$$ y = \sin \frac{1}{x} ,\ \ 0 < x \leq 1 , $$

and the interval $ I = \{ {( 0 , y ) } : {- 1 \leq y \leq 1 } \} $ gives an example of a continuum that is not locally connected (at the points of $ I $). 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 $ K $ are contained in a simple arc lying in $ K $.