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The Alexandroff-Urysohn Square and the Fixed Point Property
Fixed Point Theory and Applications volume 2009, Article number: 310832 (2009)
Abstract
Every continuous function of the Alexandroff-Urysohn Square into itself has a fixed point. This follows from G. S. Young's general theorem (1946) that establishes the fixed-point property for every arcwise connected Hausdorff space in which each monotone increasing sequence of arcs is contained in an arc. Here we give a short proof based on the structure of the Alexandroff-Urysohn Square.
Alexandroff and Urysohn [1] in Mémoire sur les espaces topologiques compacts defined a variety of important examples in general topology. The final manuscript for this classical paper was prepared in 1923 by Alexandroff shortly after the death of Urysohn. On [1, page 15], Alexandroff denoted a certain space by . While Steen and Seebach in Counterexamples in Topology [2, Example  101] refer to this space as the Alexandroff Square, we concur with Cameron [3, pages 791-792], who attributes it to Urysohn. Hence we refer to
as the Alexandroff-Urysohn Square and for convenience denote it by
. The following definition of
is given by Steen and Seebach [2, Example  101, pages 120-121]. Define
to be the closed unit square
with the topology
defined by taking as a neighborhood basis of each point (
) off the diagonal
the intersection of
with open vertical line segments centered at (
) (e.g.,
). Neighborhoods of each point
are the intersection with
of open horizontal strips less a finite number of vertical lines (e.g.,
and
). Note
is not first countable, and therefore not metrizable. However,
is a compact arcwise-connected Hausdorff space [2].
In Young's paper [4] of 1946, local connectivity is introduced on a space by a change of topology with consequent implications on generalized dendrites. A non-specialist may not notice that the fixed-point property for the Alexandroff-Urysohn Square follows from a result in Young's paper. We offer the following short proof based on the structure of the Alexandroff-Urysohn Square. The proof is direct and uses a dog-chases-rabbit argument [5, page 123–125]; first having the dog run up the diagonal, and then up (or down) a vertical fiber. The Alexandroff-Urysohn Square is a Hausdorff dendroid. For a dog-chases-rabbit argument that metric dendroids have the fixed point property, see [6], and also see [7].
Definition 1.
A set in
is an ordered segment if
is a connected vertical linear neighborhood or
is a component of the intersection of
and a horizontal strip neighborhood.
Note the relative topology induced on each ordered segment by is the Euclidean topology. Each point of
is contained in arbitrarily small ordered segments.
Let be the function defined by
. Since each neighborhood in
of a point of
is projected by
onto the complement of a finite set in
, the function
is discontinuous at each point of
.
Let be the function defined by
. Note
is continuous.
Lemma 2.
Let be a continuous function. Let
be a point of
. If
, then there is an ordered segment
containing
such that
is in one component of
.
Proof.
Suppose . We consider two cases.
Case 1.
Assume . Let
be a vertical ordered segment containing
.
Since p and f is continuous, there is a horizontal strip neighborhood H in
of p such that
and
. Let
be the
-component of
. Note
is an ordered segment containing
and
. The point
is contained in one component of
.
Case 2.
Assume . Let
be a horizontal strip neighborhood in
of
such that
and
is connected. Let
be the
-component of
. Note
is a square set with diagonal
.
Let be a horizontal strip neighborhood in
of
such that
and
. Let
be the ordered segment that is the
-component of
. Note
is a connected subset of
and
. Hence
is in one component of
. This completes the proof of our lemma.
Theorem 3.
The Alexandroff-Urysohn Square has the fixed-point property.
Proof.
Let be a continuous function. We will show there exists a point of
that is not moved by
.
Let . Note
. Let
be the least upper bound of
.
Note . To see this assume
. Then, by the lemma, there is an ordered segment
in
containing
such that
is in one component of
. However since
is the least upper bound of
, there exist points
and
in
such that
and
, a contradiction. Hence,
.
If , then
as desired.
If , then either
or
. Assume without loss of generality that
.
Let denote the interval
.
Let be the function defined by
if
and
if
.
Note is an open and closed subset of
. It follows that
is continuous. Thus,
is a retraction of
to
.
Let be the restriction of
to
. Since
is a continuous function of the interval
into itself, there is a point
such that
.
Since every point of that is sent into
by
is moved by
, it follows that
. Hence
.
References
Alexandroff PS, Urysohn P: Mémoire sur les espaces topologiques compacts. Verhan-Delingen der Koninklijke Akademie van Wetenschappen te Amsterdam 1929, 14: 1–96.
Steen LA, Seebach, JA Jr.: Counterexamples in Topology. Holt, Rinehart Winston, NY, USA; 1970:xiii+210.
Cameron DE: The Alexandroff-Sorgenfrey line. In Handbook of the History of General Topology. Volume 2. Edited by: Aull CE, Lowen R. Springer, New York, NY, USA; 1998:791–796.
Young GS Jr.: The introduction of local connectivity by change of topology. American Journal of Mathematics 1946, 68: 479–494. 10.2307/2371828
Bing RH: The elusive fixed point property. The American Mathematical Monthly 1969, 76: 119–132. 10.2307/2317258
Nadler SB Jr.: The fixed point property for continua. Aportaciones Matemáticas 2005, 30: 33–35.
Borsuk K: A theorem on fixed points. Bulletin of the Polish Academy of Sciences 1954, 2: 17–20.
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Foregger, T.H., Hagopian, C.L. & Marsh, M.M. The Alexandroff-Urysohn Square and the Fixed Point Property. Fixed Point Theory Appl 2009, 310832 (2009). https://doi.org/10.1155/2009/310832
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DOI: https://doi.org/10.1155/2009/310832