Knaster–Kuratowski fan

In <a href="/facts/Topology/2EqW47cU">topology</a>, a branch of mathematics, the Knaster–Kuratowski fan (named after Polish mathematicians <a href="/facts/Bronis%C5%82aw_Knaster/F8yQlntc">Bronisław Knaster</a> and <a href="/facts/Kazimierz_Kuratowski/8xoKvo5b">Kazimierz Kuratowski</a>) is a specific <a href="/facts/Connected_space/5CUTxfoA">connected</a> <a href="/facts/Topological_space/v2ut7CbK">topological space</a> with the property that the removal of a single point makes it <a href="/facts/Totally_disconnected/lTu5Rvyw">totally disconnected</a>. It is also known as Cantor's leaky tent or Cantor's <a href="/facts/Tipi/L2bKczgg">teepee</a> (after <a href="/facts/Georg_Cantor/k5v1DvW4">Georg Cantor</a>), depending on the presence or absence of the <a href="/facts/Apex_(geometry)/SrMxmHJk">apex</a>.
Let 
 
 
 
 C
 
 
 {\displaystyle C}
 
 be the <a href="/facts/Cantor_set/6NUcMJvN">Cantor set</a>, let 
 
 
 
 p
 
 
 {\displaystyle p}
 
 be the point 
 
 
 
 
 (
 
 
 
 
 1
 2
 
 
 
 ,
 
 
 
 1
 2
 
 
 
 
 )
 
 ∈
 
 
 R
 
 
 2
 
 
 
 
 {\displaystyle \left({\tfrac {1}{2}},{\tfrac {1}{2}}\right)\in \mathbb {R} ^{2}}
 
, and let 
 
 
 
 L
 (
 c
 )
 
 
 {\displaystyle L(c)}
 
, for 
 
 
 
 c
 ∈
 C
 
 
 {\displaystyle c\in C}
 
, denote the line segment connecting 
 
 
 
 (
 c
 ,
 0
 )
 
 
 {\displaystyle (c,0)}
 
 to 
 
 
 
 p
 
 
 {\displaystyle p}
 
. If 
 
 
 
 c
 ∈
 C
 
 
 {\displaystyle c\in C}
 
 is an endpoint of an interval deleted in the Cantor set, let 
 
 
 
 
 X
 
 c
 
 
 =
 {
 (
 x
 ,
 y
 )
 ∈
 L
 (
 c
 )
 :
 y
 ∈
 
 Q
 
 }
 
 
 {\displaystyle X_{c}=\{(x,y)\in L(c):y\in \mathbb {Q} \}}
 
; for all other points in 
 
 
 
 C
 
 
 {\displaystyle C}
 
 let 
 
 
 
 
 X
 
 c
 
 
 =
 {
 (
 x
 ,
 y
 )
 ∈
 L
 (
 c
 )
 :
 y
 ∉
 
 Q
 
 }
 
 
 {\displaystyle X_{c}=\{(x,y)\in L(c):y\notin \mathbb {Q} \}}
 
; the Knaster–Kuratowski fan is defined as 
 
 
 
 
 ⋃
 
 c
 ∈
 C
 
 
 
 X
 
 c
 
 
 
 
 {\displaystyle \bigcup _{c\in C}X_{c}}
 
 equipped with the <a href="/facts/Subspace_topology/NjU6z4Ca">subspace topology</a> inherited from the standard topology on 
 
 
 
 
 
 R
 
 
 2
 
 
 
 
 {\displaystyle \mathbb {R} ^{2}}
 
.
The fan itself is connected, but becomes totally disconnected upon the removal of 
 
 
 
 p
 
 
 {\displaystyle p}
 
.

Knaster–Kuratowski fan open-in-new

Knaster–Kuratowski fan