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A topological space  $  X $
such that for any point  $  x $
and any neighbourhood  $  O _ {x} $
of it there is a smaller connected neighbourhood  $  U _ {x} $
of  $  x $. 
Any open subset of a locally connected space is locally connected. Any connected component of a locally connected space is open-and-closed. A space  $  X $
is locally connected if and only if for any family  $  \{ A _ {t} \} $
of subsets of  $  X $,

$$ 
\partial  \cup _ { t } A _ {t}  \subset  \ 
{\cup _ { t } {\partial  A _ {t} } } bar
$$

(here  $  \partial  B $
is the boundary of  $  B $
and  $  \overline{B}\; $
is the closure of  $  B $). 
Any [[Locally path-connected space|locally path-connected space]] is locally connected. A partial converse of this assertion is the following: Any complete metric locally connected space is locally path-connected (the Mazurkiewicz–Moore–Menger theorem).

====Comments====

====References====
<table><TR><TD valign="top">[a1]</TD> <TD valign="top">  G.L. Kelley,   "General topology" , v. Nostrand  (1955)  pp. 61</TD></TR><TR><TD valign="top">[a2]</TD> <TD valign="top">  E. Čech,   "Topological spaces" , Interscience  (1966)  pp. §21B</TD></TR></table>