Plane fixed point problem

DefinitionFixed point property#✓fully proved

A subset \(K\) of the Euclidean plane \(\mathbb{R}^2\) has the fixed point property when every continuous map \(f \colon K \to K\) has a fixed point, that is, a point \(x \in K\) with \(f(x) = x\).

def HasFixedPointProperty (K : Set (EuclideanSpace ℝ (Fin 2))) : Prop :=
  ∀ f : (↥K → ↥K), Continuous f → ∃ x : ↥K, f x = x

Sources

• Open Problem Garden, node ID 37295, http://www.openproblemgarden.org/op/nonseparating_planar_continuum.

Used by 1

Used byNonseparating planar continuum conjecture

DefinitionNonseparating planar continuum conjecture#✓fully proved

The conjecture asserts that every nonempty, compact, path-connected subset \(K\) of the Euclidean plane \(\mathbb{R}^2\) whose complement \(\mathbb{R}^2 \setminus K\) is connected (that is, \(K\) does not separate the plane) has the fixed point property: every continuous map \(f \colon K \to K\) has a point \(x \in K\) with \(f(x) = x\).

def conjecture : Prop :=
  ∀ K : Set (EuclideanSpace ℝ (Fin 2)),
    IsCompact K → IsPathConnected K → IsConnected Kᶜ → HasFixedPointProperty K

Sources

• Open Problem Garden, node ID 37295, http://www.openproblemgarden.org/op/nonseparating_planar_continuum.

Uses 1 · Used by 1

Statement usesFixed point property

Used byPath-connected nonseparating plane continua have the fixed point property

TheoremPath-connected nonseparating plane continua have the fixed point property#●ready to prove

Let \(K\) be a nonempty, compact, path-connected subset of the Euclidean plane \(\mathbb{R}^2\) whose complement \(\mathbb{R}^2 \setminus K\) is connected. Then \(K\) has the fixed point property: every continuous map \(f \colon K \to K\) has a point \(x \in K\) with \(f(x) = x\).

This answers the nonseparating planar continuum conjecture in the affirmative. Hagopian (A fixed-point theorem for plane continua, Bull. AMS 77 (1971), 351-354; addendum 1972) proved that every arcwise connected nonseparating plane continuum has the fixed point property, and compact path-connected subsets of the plane are arcwise connected.

public theorem_wanted conjecture_holds : conjecture

Sources

• A. M. Blokh, R. J. Fokkink, J. C. Mayer, L. G. Oversteegen, E. D. Tymchatyn, Fixed point theorems for plane continua with applications, Mem. Amer. Math. Soc. 224 (2013), no. 1053, arXiv:1004.0214 (citing C. L. Hagopian, Bull. Amer. Math. Soc. 77 (1971), 351-354), https://arxiv.org/abs/1004.0214.

Uses 1

Statement usesNonseparating planar continuum conjecture

2 fully proved1 ready to prove

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