Knaster Kuratowski fan

Knaster-Kuratowski fan

The Knaster-Kuratowski fan is a special topological space that goes back to the mathematicians Bronisław Knaster and Kazimierz Kuratowski . The term supporter, in German subjects refers to the geometric shape as a subspace of the plane . Another name is Cantor - Teepee , which is obviously also an allusion to the geometric shape and at the same time contains a reference to the Cantor set on which the construction is based . It is a contiguous space that becomes totally incoherent after removing a point .

construction

Let it be the Cantor set, that is, the set of all points that have a decimal expansion to base 3 consisting only of the digits 0 and 2 . be the point . To everyone is the route that connects with . For be now ${\displaystyle C\subset [0,1]}$ ${\displaystyle p}$ ${\displaystyle \textstyle ({\frac {1}{2}},{\frac {1}{2}})}$ ${\displaystyle c\in C}$ ${\displaystyle {\overline {cp}}}$ ${\displaystyle c}$ ${\displaystyle p}$ ${\displaystyle c\in C}$

${\displaystyle X_{c}:=\{(x,y)\in {\overline {cp}}|\,y\in \mathbb {Q} \}}$,

if there is a terminating development consisting of the digits 0 and 2 to base 3, and ${\displaystyle c}$

${\displaystyle X_{c}:=\{(x,y)\in {\overline {cp}}|\,y\in \mathbb {R} \setminus \mathbb {Q} \}}$

for everyone else . The space ${\displaystyle c\in C}$

${\displaystyle X:=\bigcup _{c\in C}X_{c}\subset \mathbb {R} ^{2}}$

with that of induced relative topology is the Knaster-Kuratowski fan . ${\displaystyle \mathbb {R} ^{2}}$

The sub-room is called dotted Knaster-Kuratowski fan . ${\displaystyle X\setminus \{p\}}$

properties

• The Knaster-Kuratowski Fan is a separable , metric room , because it is a sub- room of . ${\displaystyle \mathbb {R} ^{2}}$
• The Knaster-Kuratowski fan is cohesive. Namely, if with disjoint and open sets and , then one of the two sets must contain. Then you can show that this amount has to be whole . ${\displaystyle X=U\cup V}$ ${\displaystyle U}$ ${\displaystyle V}$ ${\displaystyle p}$ ${\displaystyle X}$
• The dotted Knaster-Kuratowski fan is totally incoherent. This is mainly due to the fact that and for two different from in can be separated by a straight line. Between and is namely a point and the line through and the desired properties. Since each of them is totally incoherent, one can conclude that is is totally incoherent. ${\displaystyle X\setminus \{p\}}$ ${\displaystyle X_{c_{1}}\setminus \{p\}}$ ${\displaystyle X_{c_{2}}\setminus \{p\}}$ ${\displaystyle c_{1}\not =c_{2}}$ ${\displaystyle C}$ ${\displaystyle \mathbb {R} ^{2}}$ ${\displaystyle c_{1}}$ ${\displaystyle c_{2}}$ ${\displaystyle x\in [0,1]\setminus C}$ ${\displaystyle x}$ ${\displaystyle p}$ ${\displaystyle X_{c}\setminus \{p\}}$ ${\displaystyle X\setminus \{p\}}$
• The subspace is not totally separated. As is well known, totally disconnected follows from totally separated ; so here we have an example that the converse does not hold in general. ${\displaystyle X\setminus \{p\}}$

Individual evidence

1. ↑ B. Knaster, C. Kuratowski: Sur les ensembles connexes , Fundamenta Mathematicae (1921), Volume 2, pages 206-255
2. ↑ Lynn Arthur Steen, J. Arthur Seebach: Counterexamples in Topology. Springer-Verlag, 1978, ISBN 3-540-90312-7 , example 129
3. ↑ Michel Coornaert: Topological Dimension and Dynamical Systems , Springer-Verlag 2015, ISBN 978-3-319-19793-7 , sentence 5.2.2
4. ↑ Michel Coornaert: Topological Dimension and Dynamical Systems , Springer-Verlag 2015, ISBN 978-3-319-19793-7 , sentence 5.2.1
5. ↑ Michel Coornaert: Topological Dimension and Dynamical Systems , Springer-Verlag 2015, ISBN 978-3-319-19793-7 , sentence 5.2.3