View source for Kuratowski-Knaster fan

''Knaster–Kuratowski fan''

A totally disconnected set in the plane which becomes connected when just one point is added. Constructed by B. Knaster and C. Kuratowski [[#References|[1]]] as follows. Let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k056/k056030/k0560301.png" /> be the perfect [[Cantor set|Cantor set]], <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k056/k056030/k0560302.png" /> the subset of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k056/k056030/k0560303.png" /> consisting of the points <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k056/k056030/k0560304.png" /> such that, beginning from some <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k056/k056030/k0560305.png" />, the numbers <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k056/k056030/k0560306.png" /> are either all zero or all equal to 2; and let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k056/k056030/k0560307.png" /> be the set of all the other points. Now, let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k056/k056030/k0560308.png" /> be the point on the plane with coordinates <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k056/k056030/k0560309.png" />, and let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k056/k056030/k05603010.png" /> be the segment joining a variable point <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k056/k056030/k05603011.png" /> of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k056/k056030/k05603012.png" /> to the point <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k056/k056030/k05603013.png" />. Finally, let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k056/k056030/k05603014.png" /> be the set of all points of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k056/k056030/k05603015.png" /> that have rational ordinates for <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k056/k056030/k05603016.png" />, and let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k056/k056030/k05603017.png" /> be the set of all points of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k056/k056030/k05603018.png" /> that have irrational ordinates for <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k056/k056030/k05603019.png" />. Then

<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/k/k056/k056030/k05603020.png" /></td> </tr></table>

is connected, although <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k056/k056030/k05603021.png" /> is totally disconnected, so that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k056/k056030/k05603022.png" /> is a Knaster–Kuratowski fan.

====References====
<table><TR><TD valign="top">[1]</TD> <TD valign="top">  B. Knaster,   C. Kuratowski,   "Sur les ensembles connexes"  ''Fund. Math.'' , '''2'''  (1921)  pp. 206–255</TD></TR></table>