View source for Locally connected space

A topological space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l060/l060340/l0603401.png" /> such that for any point <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l060/l060340/l0603402.png" /> and any neighbourhood <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l060/l060340/l0603403.png" /> of it there is a smaller connected neighbourhood <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l060/l060340/l0603404.png" /> of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l060/l060340/l0603405.png" />. 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 <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l060/l060340/l0603406.png" /> is locally connected if and only if for any family <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l060/l060340/l0603407.png" /> of subsets of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l060/l060340/l0603408.png" />,

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l060/l060340/l0603409.png" /></td> </tr></table>

(here <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l060/l060340/l06034010.png" /> is the boundary of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l060/l060340/l06034011.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l060/l060340/l06034012.png" /> is the closure of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l060/l060340/l06034013.png" />). 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).



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====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>