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Can the Brouwer fixed point theorem be formulated for non-Hausdorff spaces ?

More particularly, is there a formulation of the Brouwer fixed point theorem which covers both the standard case of metrizable spaces, and the case of finite topological spaces ?

For example, is the following true:

Each endomorphism of a quasi-compact $T_0$ space which is an absolute extensor for normal spaces, necessarily has a fixed point.

As pointed out in comments by Michael Greinecker, (Kinoshita, 1953)[http://matwbn.icm.edu.pl/ksiazki/fm/fm40/fm4019.pdf]Kinoshita, 1953 gives a counterexample to the following which is a contractible compact subset of $\Bbb R^3$, see also (Bing,The elusive fixed point property)[http://www.jstor.org/stable/2317258]Bing,The elusive fixed point property and references in The generalization of Brouwer's fixed point theorem?.

Each endomorphism of a contractible quasi-compact $T_0$ space necessarily has a fixed point. (false!)

hence we replace "contractible" by "absolute extensor"

Note that being $T_0$ is necessary: the indiscrete space with two points is quasi-compact and contractible, yet the permutation of the two points has no fixed point. Perhaps one can relax the notion of a fixed point, e.g. by requiring that $x$ and $f(x)$ are topologically indistinguishable.

By the Brouwer theorem for finite topological spaces I mean what is implied by the Brouwer fixed point theorem for the geometric realisation of a finite topological space: namely, an endomorphism $f:K\to K$ of a finite topological space $K$ has a fixed point provided the endomorphism $|f|: |K|\to |K|$$\lvert f\rvert\colon\lvert K\rvert\to\lvert K\rvert$ of the geometric realisation of $K$ has a fixed point.

Can the Brouwer fixed point theorem be formulated for non-Hausdorff spaces ?

More particularly, is there a formulation of the Brouwer fixed point theorem which covers both the standard case of metrizable spaces, and the case of finite topological spaces ?

For example, is the following true:

Each endomorphism of a quasi-compact $T_0$ space which is an absolute extensor for normal spaces, necessarily has a fixed point.

As pointed out in comments by Michael Greinecker, (Kinoshita, 1953)[http://matwbn.icm.edu.pl/ksiazki/fm/fm40/fm4019.pdf] gives a counterexample to the following which is a contractible compact subset of $\Bbb R^3$, see also (Bing,The elusive fixed point property)[http://www.jstor.org/stable/2317258] and references in The generalization of Brouwer's fixed point theorem?.

Each endomorphism of a contractible quasi-compact $T_0$ space necessarily has a fixed point. (false!)

hence we replace "contractible" by "absolute extensor"

Note that being $T_0$ is necessary: the indiscrete space with two points is quasi-compact and contractible, yet the permutation of the two points has no fixed point. Perhaps one can relax the notion of a fixed point, e.g. by requiring that $x$ and $f(x)$ are topologically indistinguishable.

By the Brouwer theorem for finite topological spaces I mean what is implied by the Brouwer fixed point theorem for the geometric realisation of a finite topological space: namely, an endomorphism $f:K\to K$ of a finite topological space $K$ has a fixed point provided the endomorphism $|f|: |K|\to |K|$ of the geometric realisation of $K$ has a fixed point.

Can the Brouwer fixed point theorem be formulated for non-Hausdorff spaces ?

More particularly, is there a formulation of the Brouwer fixed point theorem which covers both the standard case of metrizable spaces, and the case of finite topological spaces ?

For example, is the following true:

Each endomorphism of a quasi-compact $T_0$ space which is an absolute extensor for normal spaces, necessarily has a fixed point.

As pointed out in comments by Michael Greinecker, Kinoshita, 1953 gives a counterexample to the following which is a contractible compact subset of $\Bbb R^3$, see also Bing,The elusive fixed point property and references in The generalization of Brouwer's fixed point theorem?.

Each endomorphism of a contractible quasi-compact $T_0$ space necessarily has a fixed point. (false!)

hence we replace "contractible" by "absolute extensor"

Note that being $T_0$ is necessary: the indiscrete space with two points is quasi-compact and contractible, yet the permutation of the two points has no fixed point. Perhaps one can relax the notion of a fixed point, e.g. by requiring that $x$ and $f(x)$ are topologically indistinguishable.

By the Brouwer theorem for finite topological spaces I mean what is implied by the Brouwer fixed point theorem for the geometric realisation of a finite topological space: namely, an endomorphism $f:K\to K$ of a finite topological space $K$ has a fixed point provided the endomorphism $\lvert f\rvert\colon\lvert K\rvert\to\lvert K\rvert$ of the geometric realisation of $K$ has a fixed point.

replaced "contractible" by "absolute extensor" in response to comment by
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Can the Brouwer fixed point theorem be formulated for non-Hausdorff spaces ?

More particularly, is there a formulation of the Brouwer fixed point theorem which covers both the standard case of metrizable spaces, and the case of finite topological spaces ?

For example, is the following true:

Each endomorphism of a quasi-compact $T_0$ space which is an absolute extensor for normal spaces, necessarily has a fixed point.

As pointed out in comments by Michael Greinecker, (Kinoshita, 1953)[http://matwbn.icm.edu.pl/ksiazki/fm/fm40/fm4019.pdf] gives a counterexample to the following which is a contractible compact subset of $\Bbb R^3$, see also (Bing,The elusive fixed point property)[http://www.jstor.org/stable/2317258] and references in The generalization of Brouwer's fixed point theorem?.

Each endomorphism of a contractible quasi-compact $T_0$ space necessarily has a fixed point. (false!)

hence we replace "contractible" by "absolute extensor"

Note that being $T_0$ is necessary: the indiscrete space with two points is quasi-compact and contractible, yet the permutation of the two points has no fixed point. Perhaps one can relax the notion of a fixed point, e.g. by requiring that $x$ and $f(x)$ are topologically indistinguishable.

By the Brouwer theorem for finite topological spaces I mean what is implied by the Brouwer fixed point theorem for the geometric realisation of a finite topological space: namely, an endomorphism $f:K\to K$ of a finite topological space $K$ has a fixed point provided the endomorphism $|f|: |K|\to |K|$ of the geometric realisation of $K$ has a fixed point.

Can the Brouwer fixed point theorem be formulated for non-Hausdorff spaces ?

More particularly, is there a formulation of the Brouwer fixed point theorem which covers both the standard case of metrizable spaces, and the case of finite topological spaces ?

For example, is the following true:

Each endomorphism of a contractible quasi-compact $T_0$ space necessarily has a fixed point.

Note that being $T_0$ is necessary: the indiscrete space with two points is quasi-compact and contractible, yet the permutation of the two points has no fixed point. Perhaps one can relax the notion of a fixed point, e.g. by requiring that $x$ and $f(x)$ are topologically indistinguishable.

By the Brouwer theorem for finite topological spaces I mean what is implied by the Brouwer fixed point theorem for the geometric realisation of a finite topological space: namely, an endomorphism $f:K\to K$ of a finite topological space $K$ has a fixed point provided the endomorphism $|f|: |K|\to |K|$ of the geometric realisation of $K$ has a fixed point.

Can the Brouwer fixed point theorem be formulated for non-Hausdorff spaces ?

More particularly, is there a formulation of the Brouwer fixed point theorem which covers both the standard case of metrizable spaces, and the case of finite topological spaces ?

For example, is the following true:

Each endomorphism of a quasi-compact $T_0$ space which is an absolute extensor for normal spaces, necessarily has a fixed point.

As pointed out in comments by Michael Greinecker, (Kinoshita, 1953)[http://matwbn.icm.edu.pl/ksiazki/fm/fm40/fm4019.pdf] gives a counterexample to the following which is a contractible compact subset of $\Bbb R^3$, see also (Bing,The elusive fixed point property)[http://www.jstor.org/stable/2317258] and references in The generalization of Brouwer's fixed point theorem?.

Each endomorphism of a contractible quasi-compact $T_0$ space necessarily has a fixed point. (false!)

hence we replace "contractible" by "absolute extensor"

Note that being $T_0$ is necessary: the indiscrete space with two points is quasi-compact and contractible, yet the permutation of the two points has no fixed point. Perhaps one can relax the notion of a fixed point, e.g. by requiring that $x$ and $f(x)$ are topologically indistinguishable.

By the Brouwer theorem for finite topological spaces I mean what is implied by the Brouwer fixed point theorem for the geometric realisation of a finite topological space: namely, an endomorphism $f:K\to K$ of a finite topological space $K$ has a fixed point provided the endomorphism $|f|: |K|\to |K|$ of the geometric realisation of $K$ has a fixed point.

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user494312
  • 513
  • 2
  • 6

Brouwer fixed point theorem for non-Hausdorff spaces

Can the Brouwer fixed point theorem be formulated for non-Hausdorff spaces ?

More particularly, is there a formulation of the Brouwer fixed point theorem which covers both the standard case of metrizable spaces, and the case of finite topological spaces ?

For example, is the following true:

Each endomorphism of a contractible quasi-compact $T_0$ space necessarily has a fixed point.

Note that being $T_0$ is necessary: the indiscrete space with two points is quasi-compact and contractible, yet the permutation of the two points has no fixed point. Perhaps one can relax the notion of a fixed point, e.g. by requiring that $x$ and $f(x)$ are topologically indistinguishable.

By the Brouwer theorem for finite topological spaces I mean what is implied by the Brouwer fixed point theorem for the geometric realisation of a finite topological space: namely, an endomorphism $f:K\to K$ of a finite topological space $K$ has a fixed point provided the endomorphism $|f|: |K|\to |K|$ of the geometric realisation of $K$ has a fixed point.