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@ Appl. Gen. Topol. 16, no. 1(2015), 31-36doi:10.4995/agt.2015.2305
c© AGT, UPV, 2015

On fixed-point free selections and

multivalued maps on R

Dennis K. Burke
a
and Raushan Z. Buzyakova

b

a
Department of Mathematics, Miami University, Oxford, OH, U.S.A. (burkedk@miamioh.edu)

b
U.S.A. (Raushan Buzyakova@yahoo.com)

Abstract

We study fixed-point free multivalued selfmaps on R as well as con-

tinuous fixed-point free selection of multivalued selfmaps on R. We

justify necessary conditions in some earlier results and motivate some

new interesting questions.

2010 MSC: 54H25; 58C30; 54B20.

Keywords: fixed point; hyperspace; multivalued function.

1. Introduction

One of the classical theorems of de Bruijn-Erdös [2] states that if f is a fixed-
point free multivalued selfmap on a set X and the sizes of f(x)’s are bounded
by a fixed natural number N, then X can be covered by finitely many sets
each of which misses its image. A more precise statement is as follows: Let
P(X) be the set of all non-empty subsets of X and let f : X → P(X) be a map
with the property that x 6∈ f(x). If there exists a natural number k such that
|f(x)| ≤ k for all x ∈ X, then there exists a finite cover F of X such that F
misses

⋃
{f(x) : x ∈ F} for each F ∈ F.

The sets Fs in the above statement are called colors of f and the family
F a coloring of f. In the context of a map (not necessarily continuous) on
a topological space, a color of the map is traditionally a closed subset of the
space. To be more precise, given a multivalued selfmap f on a topological space
X, a closed F ⊂ X is a color of f if f(x) misses F for each x ∈ F . A finite cover
of X by colors of f is called a coloring of f. If a coloring of f exists, f is called

Received 14 April 2014 – Accepted 22 September 2014

http://dx.doi.org/10.4995/agt.2015.2305


D. K. Burke and R. Z. Buzyakova

colorable. The condition of closed-ness of a color in the topological context is
justified by the fact that for a space X from a sufficiently wide class, colorability
of a continuous selfmap f on X by closed colors is equivalent to the existence
of a continuous fixed-point free extension of f to some compactifcation of X.
The notion of colorings for continuous single-valued selfmaps was introduced
in [1]. A nice review of results about this topic for single-valued maps is given
in [6].

It is proved in [3] that a fixed-point free continuous selfmap on R is colorable
if the size of f(x) is at most N, where N is a fixed positive number determined
by f. This result is a natural topologization of the mentioned de Bruijn-Erdös
theorem. In the first part of this paper we will consider a few routes to extend
this result to certain fixed-point free multivalued selfmaps on R with compact
images. For this mission, it is natural to consider the following restrictions on
f:

(1) There exists M > 0 such that sup{d(x, y) : y ∈ f(x)} < M for all x;
and/or

(2) There exists m > 0 such that d(x, f(x)) > m for all x.

We will show that (1) alone or (2) alone leads to an example. At the same
time if a multivalued selfmap on R (not necessarily continuous) satisfies both
(1) and (2), then it is colorable (Theorem 2.1).

It is clear that if f has a fixed point then it is not colorable. If, however, f
is multivalued, it may have a fixed-point free continuous selection. It is shown
in [4] that a continuous fixed-point free map on Rn is always colorable. This
observation inspired the second part of the paper, in which we discuss possible
conditions that guarantee the existence of continuous fixed-point free selections
of multivalued selfmaps on R.

By exp R we denote the space of all non-empty closed subsets of R endowed
with the Vietoris topology. By expK(R) we denote the space of all non-empty
compact subsets of R with the topology inherited from exp R. A standard
neighborhood in exp R is in form

〈U1, ..., Um〉 = {A ∈ exp R : A ⊂ U1∪...∪Um and Ui∩A 6= ∅ for all i = 1, ..., m},

where U1, ..., Um are open sets of R. Throughout the paper when we say that
f is a multivalued selfmap on R, we mean that f is a map from R to exp R .
We reserve the letter ”d” to measure distances in R.

In notations and terminology, we will follow [5].

2. Continuous Fixed-Point Free Multivalued Selfmaps on R

In this section, we will consider mulitivalued fixed-point free selfmaps on
R with compact images. Our goal is to find a large subclass of such maps
whose members are colorable. It is natural to assume that a more ”volatile”
graph is less likely to be colorable while a fixed-point fee map with a ”more
orderly conduct” should naturally be colorable. To continue our discussion,
it is appropriate at this point to define how exactly we measure the volatility

c© AGT, UPV, 2015 Appl. Gen. Topol. 16, no. 1 32



On fixed-point free selections and multivalued maps on R

of a map f. For this purpose, we introduce the following function-determined
constants:

SQ(f) = inf{d(x, y) : x ∈ R, y ∈ f(x)}, ST (f) = sup{d(x, y) : x ∈ R, y ∈ f(x)}.

One can view SQ(f) as the squeeze measure of f, which is, simply the
greatest lower bound of the distances between points and their images.

Similarly, the constant ST (f) can be viewed as a stretch measure. The
authors think of a fixed-point free map f as being volatile if its squeeze measure
SQ(f) is 0 or its stretch measure ST (f) is ∞. Thus, we may think of f is not
volatile if both SQ(f) and ST (f) are positive real numbers. In this section
we will show that any non-volatile fixed-point free (not necessarily continuous)
multivalued map on R is colorable. We will also show that for either of the two
measures of volatility, there exists an example of a non-colorable continuous
fixed-point free multivalued selfmap on R that obeys the given measure and
violates the other. We start with our affirmative result.

Theorem 2.1. Let f : R → exp(R) be a map and m, M positive numbers.
Suppose that m ≤ d(x, y) ≤ M for every x ∈ R and y ∈ f(x). Then f is
colorable.

Proof. Put I = {[m
2
· n, m

2
· (n + 1)] : n is an integer}. Clearly I is a cover

of R. Let K be an integer such that K · m
2

> M. We define our colors
{Ai : i = 0, ..., K} as follows.

Ai is the union of all elements of I that can be obtained by shifting [
m
2
·i, m

2
·i+

m
2
] a multiple of K + 1 units to the right or to the left. That is, Ai is the union

of the largest subfamily of I that contains [m
2
· i, m

2
· i + m

2
] and has neighbors

at distance exactly m
2
· K from each other.

It is clear that each Ai is closed and {Ai : i = 0, ..., K} is a cover of R. Let
us show that Ai is a color. Note that Ai is the union of a disjoint collection of
segments of length m/2. Since d(x, f(x)) ≥ m for each x, we conclude that the
image of each segment I of this collection misses I. Also, since the distance
between any two distinct segments I and J of this collection is m

2
· K > M,

the image of I misses J. Therefore, the image of Ai misses Ai. Hence, Ai is a
color. �

Recall that given a colorable multivalued map f on a topological space, the
chromatic number of f is defined as the smallest number of colors necessary to
color f. Observe that the argument of Theorem 2.1 implies that the chromatic
numbers does not exceed ⌈2M

m
⌉ + 1.

We next show that either part (alone) of the inequality in the hypothesis
of Theorem 2.1 is not sufficient for the desired conclusion. We first describe
an example that shows that positiveness of SQ alone does not guarantee col-
orability.

Example 2.2. There exists a continuous fixed-point free non-colorable map
from R to expK(R) such that SQ(f) > 0, ST (f) = ∞, and the image of each
point is a segment.

c© AGT, UPV, 2015 Appl. Gen. Topol. 16, no. 1 33



D. K. Burke and R. Z. Buzyakova

Construction. For each x ≤ −1, put f(x) = [x + 1/2, 0]. For each x ≥ −1,
put f(x) = [x + 1/2, x + 1]. This map is clearly continuous, fixed-point free,
SQ(f) = 1/2, and ST (f) = ∞. Let us show that f is not colorable. Since
n ∈ f(m) for every pair of negative integers n, m with n > m, we conclude that
n and m have different colors. Therefore, −1, −2, −3, ... must have mutually
distinct colors. �

Next, we will show that finiteness of ST alone is not sufficient for colorability.
Lemma 2.3 and Example 2.4 can serve as a manual for constructing examples
with the desired properties. We will then demonstrate the manual at work.
We start with the following technical statement.

Lemma 2.3. Let a map f : R → exp(R), a ∈ R, and N ∈ ω \ {0, 1} satisfy the
following requirement:

f(a +
m

N
) = [a +

m + 1

N
, a + 1 +

m + 1

N
] for each m = 0, ..., N − 1

Then f cannot be colored by fewer than N colors.

Proof. If f is not colorable, then we are done. Otherwise, fix a coloring of f.
If m > k, then a + m

N
≥ a + k+1

N
. If m ≤ N − 1 then a + m

N
≤ a + 1 + 1

N
.

Therefore,

(P) a + m
N

∈ f(a + k
N
) for k < m ≤ N − 1

Property (P) implies that a, a+ 1
N
, ..., a+

(N−1)+1
N

must have mutually distinct
colors. �

We are now ready to construct an example showing that finiteness of SQ
alone is not sufficient for colorability of a fixed-point free continuous map from
R to expK(R).

Example 2.4. There exists a fixed-point free non-colorable continuous map
from R into expK(R) such that ST (f) is finite, SQ(f) = 0, and the image of
each point is a unit segment.

Construction. We state the next claim for further reference and it is a direct
consequence of continuity of a composition of continuous functions.

Claim. Let h : R → R and f : R → exp(R) be maps such that f(x) =
[h(x), h(x) + 1] for each x ∈ R. If h is continuous, then f is continuous too.

Fix a sequence of real numbers a2, ..., an, ... such that

(A) an+1 − an > 5, for each n

For each n = 2, 3... put

(B) g(an) = an + 1/n, g(an +
1
n
) = an +

2
n
, ..., g(an +

n−1
n

) = an + 1

Observe that x < g(x) for all x in {an + m/n : m = 0, ..., n − 1}, where
n ∈ ω \ {0, 1}. Moreover,

(C) g(an), g(an +
1
n
), ..., g(an +

(n−1)
n

) fall in [an, an + 5) ⊂ [an, an+1).

c© AGT, UPV, 2015 Appl. Gen. Topol. 16, no. 1 34



On fixed-point free selections and multivalued maps on R

Now applying (A), we conclude that x < g(x) for all x in S = {an +
m
n

: n =
2, 3... and m = 0, 1, ..., n − 1}. Since S is closed, there exists a continuous
extension h : R → R of g such that x < h(x) for all x ∈ R.

Define f : R → expK(R) by letting f(x) = [h(x), h(x) + 1]. By Claim, f is
continuous. By (B) and Lemma 2.3, f is not colorable. Since x < h(x) for all
x in the domain, f is fixed-point free. �

One can use the argument of the example and the accompanying lemma to
create well-defined maps with the desired properties. In particular, the referee
of this paper suggested the map f : R → expK(R) defined as follows:

f(x) = [x +
1

x
, x +

1

x
+ 1] if x ≥ 1 and f(x) = [x + 1, x + 2] otherwise.

Following the argument of our example and the lemma, one can see that this
map has the desired properties.

We have now shown that there are examples of non-colorable continuous
maps from R to expK R that have positive squeeze component and ones that
have finite stretch component.

3. Continuous Fixed-Point Free Selections

We start this section with a simple observation that motivates this section’s
discussion.

Proposition 3.1. A continuous map f : R → expK(R) has a fixed-point free
continuous selection if and only if {x ∈ R : x = max f(x)} or {x ∈ R : x =
min f(x)} is empty.

Proof. To prove necessity, let sf be a continuous fixed-point free selection.
Then the graph of sf is completely above the diagonal or completely below
the diagonal in R2. This means that sf (x) > x for all x or sf (x) < x for all
x. Assume that sf (x) > x for all x. Then x < max f(x) for all x. Hence
{x ∈ R : x = max f(x)} = ∅. Similarly, {x ∈ R : x = min f(x)} = ∅ if
sf (x) < x for all x.

To prove sufficiency, assume that {x ∈ R : x = max f(x)} = ∅. The map sf
defined by letting sf (x) = max f(x) is then fixed-point free and is known to be
continuous. �

An argument similar to the one in Proposition 3.1 implies the following
statement.

Proposition 3.2. Let X be a closed subset of R and f : X → expK(X)
be continuous. Then f has a fixed-point free continuous selection iff the sets
{x ∈ R : x = max f(x)} and {x ∈ R : x = min f(x)} are separated by clopen
neighborhoods.

Note that the statement of Proposition 3.1 is no longer true if we replace
expK(R) by exp(R). This is shown in the next example.

c© AGT, UPV, 2015 Appl. Gen. Topol. 16, no. 1 35



D. K. Burke and R. Z. Buzyakova

Example 3.3. There exists a continuous map f : R → exp(R) such that the
set {x ∈ R : x = max f(x)} is empty but every continuous selection of f has a
fixed point.

Construction. For n ∈ N, let Fn = ((−∞, n]) × {n}) ∪ {(x, 2n − x) : n ≤
x ≤ 2n} ∪ [2n, ∞) × {0} and E =

⋃
n∈N

Fn. Define f : R → exp(R) by
f(x) = {y : (x, y) ∈ E} = Ex (vertical slice at x.) The graph of any continuous
selection would have to ”follow” some Fn and hence would have n as a fixed
point. �

Our observation prompts the following natural questions.

Question 3.4. Does Proposition 3.1 hold of we replace expK(R) by {A ∈
exp(R) : A is connected} ?

In general, it would be interesting to indentify conditions under which a
continuous multivalued map on Rn with compact images has a fixed-point free
continuous selection. It would be also interesting to consider the n-dim versions
of Proposition 3.2 and Question 3.4. In addition to continuity we can require
our selection to have some other natural properties.

Question 3.5. Suppose that f is a continuous multivalued map on R such
that f(x) is a closed non-trivial segment and f has a continuous fixed-point
free selection. Does f have a differentiable fixed-point free selection? Does f
have a continuous fixed-point free selection differentiable at at least one point?

What if we replace R with Rn and ”closed segment” with ”closed ball”?

Note that ”non-triviality” of images is important since any continuous non-
differentiable (or nowhere differentiable) map with graph strictly above the
line y = x would give a trivial example.

Acknowledgements. The authors would like to thank the referee for many

valuable remarks, corrections, and references.

References

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Schnirelmann, Topology 35 (1996), 1051–1056.

[2] N. G. de Bruijn and P. Erdös, A color problem for infinite graphs and a problem in the
theory of relations, Indagationes Math. 13 (1951), 371–373.

[3] R. Buzyakova, On multivalued fixed-point free maps on Rn, Proc. Amer. Math. Soc.
140 (2012), 2929–2936.

[4] R. Buzyakova and A. Chigogidze, Fixed-point free maps of Euclidean spaces, Funda-
menta Mathematicae, 212 (2011), 1–16

[5] R. Engelking, General Topology, PWN, Warszawa, 1977.
[6] J. van Mill, The infinite-dimensional topology of function spaces, Elsevier, Amsterdam,

2001.

c© AGT, UPV, 2015 Appl. Gen. Topol. 16, no. 1 36