# 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 $K$.