Math 2.0

General topology Continua and dimension theory Counterexample

Knaster–Kuratowski fan

Builds on Cantor setConnected spaceTotally disconnected space

Counterexample

Let CC be the Cantor set on the segment [0,1]×{0}[0, 1] \times \set{0} of the plane, and let p=(1/2,1/2)p = (1/2, 1/2). Let E⊆CE \subseteq C be the countable set of end points of the open intervals removed in the construction of CC. For c∈Cc \in C, let L(c)L(c) be the closed segment from (c,0)(c, 0) to pp. Let XcX_c be the set of points of L(c)L(c) whose second coordinate is rational if c∈Ec \in E, and irrational if c∈C∖Ec \in C \setminus E. The Knaster–Kuratowski fan is F=⋃c∈CXc∪{p}F = \bigcup_{c \in C} X_c \cup \set{p}, with the topology of a subspace of the plane. Then FF is connected, but F∖{p}F \setminus \set{p} is totally disconnected. So removing one point from a connected space can leave no connected subset with more than one point.

Remarks

For two points c<c′c < c' of CC, pick tt in a removed interval between them. The line through pp and (t,0)(t, 0) misses F∖{p}F \setminus \set{p}, and the two open half-planes it bounds split F∖{p}F \setminus \set{p} into two open pieces. So a connected subset of F∖{p}F \setminus \set{p} lies in a single set XcX_c. Each XcX_c is homeomorphic to the rational or to the irrational points of an interval, which is totally disconnected. The proof that FF is connected is harder; it uses the Baire category theorem on the Cantor set. The point pp is called a dispersion point of FF. The space is also called Cantor's leaky tent or Cantor's teepee.

Reading path

To understand this page from the ground up, read these 4 pages in order. The foundations come first.

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Cantor set, example, no evidence levelCantor setexampleConnected space, concept, no evidence levelConnected spaceconceptTotally disconnected space, concept, no evidence levelTotally disconnected …conceptKnaster–Kuratowski fan, counterexample, no evidence levelKnaster–Kuratowski fancounterexample
Each page uses the pages above it. The colour of the bar on a box shows how strong the evidence is, and the box also gives it as text. The list has the same pages.
  • no evidence level
  1. Cantor set example
  2. Connected space concept
  3. Totally disconnected space concept
  4. Knaster–Kuratowski fan counterexample

Further reading

  1. Web https://en.wikipedia.org/wiki/Knaster%E2%80%93Kuratowski_fan
About this page
Tag3DRNRZEW9D
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Kindcounterexample
WritingW0 stub
MSC54G20, 54D05
UsesCantor set, Connected space, Totally disconnected space
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Changed2 November 2026

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