Download equidistant sets and their connectivity properties

PROCEEDINGS OF THE
AMERICAN MATHEMATICAL SOCIETY
Volume 47, Number 2, February
1975
EQUIDISTANT SETS AND THEIR CONNECTIVITY PROPERTIES
J. B. WILKER
ABSTRACT.
If A and
the set of points
set
determined
and
B are
set
determined
1.
by
line
by
points
is that
other
A and
B is connected.
is a line,
of hyperbolae
from two nested
set
of points
branch
around
the smaller
further
about
These
the properties
it is shown
then
the set
It is less
the
of sets
that
with
known
is an ellipse
foci at their
A
equidistant
from a
that
definitions.
examples
which
if
equidistant
with
can be realised
set
of
foci at their
centres
prompt
ellipses
The
from two dis joint disks
classical
(X, d),
equidistant
equidistant
of points
well
analogous
circles
equidistant
of an hyperbola
disk.
results,
plane
admit
space
the
22-space,
and the set of points
not on it is a parabola.
equidistant
The
Among
In the Euclidean
branches
centres.
of a metric
B) is called
X is Euclidean
points
and a point
B.
subsets
d(x, A) = d(x,
and
Introduction.
and single
sizes
A and
connected
from two distinct
B are nonvoid
x e X for which
of different
which
opens
us to inquire
as equidistant
sets.
The
most general
a metric
space
context
(X, d).
then the distance
in which
this
If A is a nonvoid subset
from
x to A is defined
dix, A) = inf\dix,
If A and
B are both nonvoid
determined
study
subsets
by A and ß is defined
is meaningful
is that
of
of A', and x is a point of X,
to be
a): a £ A \.
of X then
the equidistant
set
to be
U = r5| = {x: dix, A) = dix, B)\.
This
notation
admits
convenient
generalization
Received
by the editors September
AMS (A/05) subject classifications
to ÍA < B\ = \x: dix,
17, 1973.
(1970). Primary
A) <
50B99; Secondary
54E35,
55B10.
Key words
connected
set,
Research
A8100.
discussions
and phrases.
Baire
Equidistant
category,
supported
in part
set,
Mayer-Vietoris
by National
metric
Research
The author would like to thank E. Barbeau
of this
space,
Euclidean
72-space
sequence.
Council
of Canada
Grant
and T. Bloom for helpful
material.
Copyright © 1975. American Mathematical Society
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446
EQUIDISTANT SETS: CONNECTIVITY PROPERTIES
447
dix, B)\ and ÍA < B\ = \x: dix, A) < dix, B)\.
The properties
metric
space
are discussed
specialized
their
which
72-space
can be established
section.
Then
in a general
the metric
and a more detailed
analysis
space
is
is made
of
properties.
Equidistant
the closure
sets
in the next
to Euclidean
topological
2.
of equidistant
sets
of a set
in metric
A is just
spaces.
A = \x:dix,
In terms
of the point-set
A) = 0}.
It follows
dix, A) = a\x, A) and, therefore,
that \A = B} = ,A = BÎ. In general,
IA = ß! 0 A n B because
points
A u B = X, then
distance
these
have distance
ÍA = ßl = A n B because
to one set and zero distance
distance,
easily
that
zero to both sets.
other points
to the other.
If
have a positive
If A = B, then
ÍA = ß! = X.
The function
therefore
dA : X —>R defined by dAix) = dix, A) is Lipschitz
continuous.
respectively,
The sets
the inverse
continuous
function
\A = B}, \A < B\ and
images
dA — d
it is not generally
true that
nonvoid
closed
A n X = A.
set being
1.
A.
space
to show that this
set.
of {A < B\.
but, conversely,
This
any
is true because
to the possibility
X is connected
if and only
(A = X} =
of the void
if equidistant
X = A u B for some pair of nonvoid
subsets,
cannot
then
happen
when
their
ÍA < B\ is nonvoid
intersection
.
X is connected.
X = ÍA < B! U\B < A\ is the union of nonvoid
if X is connected,
\A < B] ate
void.
B are nonvoid
Thus
closed,
sets A and B. Then {A = ßS = AOß=0
It remains
A and
and
the
set.
If X is not connected,
closed disjoint
{A = ß!
of X is related
The metric
in X are never
(- 00, 0) under
{A < B\ = ,A < B\ u U = ßi, but
always
A is an equidistant
an equidistant
Proof.
sets
The connectivity
Theorem
sets
set
that
Trivially,
i A = ß} is the boundary
Not only are equidistant
{A < B\ ate,
of 1 OÍ, (- °°, O] and
. It follows
closed while ÍA < B\ is open.
and
because
closed
If
it contains
sets,
and
\A < B\ Ci ÍB < A! = ¡A = Bj must be
nonvoid.
3.
Equidistant
flat in Euclidean
sets
72-space,
E may be determined
E.
It follows
that
in Euclidean
the distance
from distances
if A and
72-space.
If E is an 772-dimensional
from a general
measured
B are subsets
point to a point
in E and perpendicular
of E, then
based on \A = BÎ n E.
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of
to
\A = B\ is a cylinder
448
J. B. WILKER
If A is nonvoid
closed
dix, a A tot some point
radius
S + 1 about
infWx,
in A.
x meets
72-space,
For if dix,
then
dix, A) =
A) = 8, the closed
A in a compact
a): a e C\ is realized
mental
set
as a minimum.
C, and
This
dix,
useful
ball
of
A) =
remark
is instru-
in proving
Theorem
such
set in Euclidean
a
that
2.
// A and
A n B = 0,
B are nonvoid
subsets
then \A = B\ has void
of Euclidean
interior
n-space
and is the common
boundary of \A < B\ and {B < A\.
Proof.
Let
theorem.
the conditions
of the
Then die, A) = die, B) = 8 > 0. The open ball of radius
8 about
e contains
least
no points
two distinct
open radial
ball
e e \A = B\ where
of A or ß while
points
segment
of radius
8.
A and
a
ß satisfy
its bounding
e A and b
£ B.
ie, a A has distance
about
x meets
sphere
If a point
5. < 8 from a.,
A U ß at the single
that ie, aQ] C ÍA < B\, and similarly
contains
at
x on the half
then
point
the closed
a . It follows
that (e, b ] C \B < A). This gives the
theorem.
As an application
infinite
family
A .ii
A .O A . = 0
numbers
fact
of the theorem,
£ I) of nonvoid
fot i 4 j.
shows
that
occurs
these
We are
equidistant
prompted
sets
that
this remains
connectivity.
Theorem
to ask
in some
dense
for a complete
are determined
after
an open question,
3.
In Euclidean
nonvoid
of two parallel
set,
for this
the
surprising
\A . = A ,J. But the
and so a countable
description
the fashion
union
below
of the
of Theorem
and so we return
stated
2.
How-
now to the issue
of
do not make use of the
B is a subset
into which
a divides
Theorem
4.
n-space
if A = \a\
\A = B\
is either
The second
case
then
hyperplanes.
a £ B and
then
The reason
are nowhere
satisfy
An B=0 .
an arbitrary
sets,
which
x in the 72-space,
category.
The main theorems
hypothesis
or denumerably
of 72-space
only if x lies
sets
Baire
a finite
points
dix, A .) (í e /) are all different.
of them is only of first
ever,
subsets
Then for "most"
is that an equality
theorem
consider
of a line through
arises
a which
if and only
meets
and
B
or the union
if
both of the rays
it.
In Euclidean
\A = Bj
is a singleton
connected
n-space
if A and
B are nonvoid
is connected.
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connected
EQUIDISTANT SETS: CONNECTIVITY PROPERTIES
It is not true that
connected.
if A and
An amusing
locked combs.
B are path
then
IA = B] is path
in the plane
is given
by two inter-
-1 +l/n<y
< 1, re=l, 3, 5, •••!
y): %= l/n, -l<y<l-l/n,
n = 2, 4, 6, • • •!.
counterexample
connected
449
Let
A = \ix, 1): x> OSu \ix, y): x=l/n,
and
B = \(x, -1): *> 0|u(U
Then
S¡4 = ßi
the graph
is the closed
of y = sin l/x,
halfplane,
x < 0, together
with
x> 0, but made up of segments
a curve
of straight
resembling
lines
and
parabolas.
Before
analogues
starting
break
to prove
down when
72-space to the circle
the equidistant
set
will indicate,
Proof
nonvoid
1.
Proof.
The set
\e
still
In this
stands
connected.
if the circle
section
in the following
A = \a\
is a singleton
\{a\ < ß!
is convex
and therefore
\{a\ < B\ is the intersection
of the closed
hálfspaces
If a £ B, dix, B) < dix, a) and so \\a\ = ß! = \\a}.< B\.
If a £ B, the equidistant
from Lemma
For the rest
Let
of the proof of Theorem
R denote
an arbitrary
HÍR.) the open halfspace
point
We shall
\\a\
= B\
is connected.
The
closed
bordering
set exactly
3 it is possible
ray issuing
to assume
a with R as inward
when
that
from the point
pointing
HÍR.) meets
a,
normal.
B and then
in
eiR) 4 a.
prove
of a sphere
(Lemma
set
1.
The ray R meets the equidistant
equidistant
B
with b £~B-\a\.
follows
topology
and
connected.
Proof.
a single
sections.
of 72-space.
2.
a é B.
is
: 0 < 6 < 377/2!.
Lemma
result
Also
subset
As the proofs
is not simply
3 and 4 are developed
3.
from Euclidean
is not connected.
the counterexample
subset
The set
is changed
how their
Here, if A = ! l! and B = {- l!,
is that the circle
connected
of Theorem
Lemma
\\a\<\b\\
and
because
of Theorems
is an arbitrary
space
{A = B\ = \i, - i\ which
by its simply
4.
the metric
part of the problem
The proofs
3 and 4, let us notice
\eiQ: 0 < 6 < 2n\.
But more is at stake
replaced
Theorems
(Lemma
about
3) that
a, then
4) for dimension
set,
then
the domain
if the rays
from a ate given
the mapping
72> 2 if every
of this
the
R —>e(R) is continuous.
line through
mapping
a meets
is connected.
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the
These
two
450
j. B. WILKER
results
prove
that
the equidistant
On the other hand,
equidistant
through
set,
then the set
a perpendicular
on \\a\ = ßi
equidistant
set
flats
where
a.
planes
\\a\
depending
B meets
Proof.
Let R„ meet the equidistant
The mapping
bA for
Since
ball.
by the argument
cylinder.
of a line
Al-
E
through
or a pair of parallel
hyperinto
a 4 B there
on rays
is a closed
it contains
provides
is continuous.
setat
e(ßj.
Then the hyperplane
eiR)
is convex,
The cone
sequence
in a
just one or both of the rays
R —'eiR)
b. £ B.
bound for points
< ß!
based
E..
3.
outer
E
(72- l)-
m >2
B is a subset
hyperplane
Lemma
dieiRA,
\\a\
that
the
flat
chosen
at dimension
set is guaranteed
a single
on whether
which a divides
An inductively
\{a\ = B\ is a connected
may prove
= BÎ is either
misses
set is a cylinder
if and only if this
equidistant
Then
a which
(72 - l)-dimensional
the equidistant
may terminate
paragraph.
the induction
Then
Thus
is connected.
a connected
of the preceding
ternatively
B must be in the
to L.
and orthogonal
situation
set is connected.
is a line L through
C[ E and will be connected
dimensional
of lines
if there
R neat
an inner
Sí«! = \bA\ provides an
a lying
of tangents
bound
dieiR A, B) =
RQ.
ball about
the cone
Let
for points
in }(a|.<Bi.
from
eiR)
Since
e(#0)
to this
on rays
R neat
R0.
Lemma
equidistant
4.
set,
Proof.
set.
set
then
If R.
angle 6 <n.
HÍR2),
In dimension
the set
and
of all rays
which
meet
it fails
to meet
lie on an arc,
B, and consequently
circle
and those
so may be joined
which
meets
to
do meet
set they meet at an
If Euclidean
circular
then
closed,
but is it connected?
The remarks
about
spaces
show
that with
answer
must
be yes
special
case
conditions
4 is true.
by allowing
set,
every
If
of the same type.
= A U B, where A
A O B is nonvoid
equidistant
sets
and
in metric
\A = B! = A O B, so the
The next
us to replace
arc.
other ray which
arc of rays
72-space E
and connected,
the preceding
to meet the equidistant
it on the complementary
closed
if Theorem
HÍR) C HÍR ) U
fail to meet the equidistant
the equidistant
RQ by a great
the
it is connected.
R fails
the rays which
5. Proof of Theorem 4.
B are nonvoid
4 to this
a meets
R2 fail to meet the equidistant
R~ is a fixed ray which
and
line through
If R is a ray in the angle 6, then because
Thus on any great
does
n > 2, if every
lemma
given
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reduces
connected
Theorem
sets
A and
EQUIDISTANT SETS: CONNECTIVITY PROPERTIES
B by the sets
ÍA < ß! and Iß < A\ from which \A - B\ may be determined
as \A <B\ n\B
Lemma
then
451
5.
<A\.
// A and
B are nonvoid
subsets
of E
and
A is connected,
\A < B\ is connected.
Proof.
Because
point-set
is the union
of the sets
connected.
Since
distances
\\a\ < ß!
with
A is connected
a £ A.
in closed
By Lemma
A in the union and deduce
standard
result
that
sets,
1, \\a\
and \\a\ < B\ n A D \a\,
include
A result
are realized
¡A < B\
< B\ is
it is possible
\A < B\ is connected
to
from the
on unions.
analogous
to the one required
to complete
the proof
of Theorem
4 is provided by
Lemma
connected,
Proof.
6.
If' E 72 = Y u Z where
then
Y Pi Z is connected.
Since
homological
V and
modules
-iHjiE)
Since
E
Z are open,
is exact.
reduces
its first
Z are nonvoid,
the Mayer-Vietoris
The tail
-* H0iY n Z)^
is contractible,
sequence
Y and
of this
open
and
c
sequence
sequence
of
is
II0iY) © //Q(Z) — H A/E ) - 0.
homology
group
is trivial
and the
to
0 -* H0iY nZ) -» H0iY) ® H0iZ) -. HQiE) -. 0.
Open connected
H.
counts
subsets
the number
of E
are path
of these
connected,
components.
and the dimension
It follows
that
of
YO Z
is path
closed
and
connected.
The proof of Theorem
Lemma
connected,
Proof.
disjoint
>
7.
If' E 72 = A U B, where
then
A C\ B is connected.,
sets
(7. such
C, 1 and
that
Cv2
with
A and
If A n B is not connected,
closed
open sets
4 is completed
Since
then
V.C
in C 2Ai=
I,' 2). Then define
(/. which
can be written
A n B = Cj U C2 for nonvoid
E 72 is normal,
C. C U. (z = 1, 2).
open sets
B are nonvoid,
Replace
there
these
as the union
are disjoint
'
open sets
of open balls
V = V,12U V, and write
by
centred
Y = A U V and
Z = B u V.
Since
together
AnßCV,
with certain
Y=ß
open balls
uV
and is open.
meeting
Since
V is equal to A
A, it is connected.
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Similarly
Z
452
J. B. WILKER
is open and connected.
By Lemma
Vj U V2 is the union
completes
6.
Addendum.
H. Bell
proof of Lemma
dix, fix))
and remark
\A < B\.
This
S. Ferry
of E2 then
that
approach
has
H. Bell
show
6, Y n Z = V is connected.
disjoint
open sets.
This
But
V =
contradiction
the proof.
alternative
In [l]
of nonvoid
{A = ß!
if x e\A
drawn
that
the distance
define
result
independently
f: E
—> A such
with
dA: E
presents
in E,.
[x, fix)] C
of related
closed
an
dix, A) =
of Theorem
to a number
He also
suggested
that
segment
the proof
B are disjoint
is false
function
have
< B\ then the closed
my attention
if A and
is a 1-manifold.
that the analogous
investigate
They
may be compared
kindly
proves
and S. K. Kaul
5.
2.
references.
connected
subsets
a counterexample
In [2]—[4]
other
—» R to see when
to
authors
its level sets
are
in - l)-manifolds.
REFERENCES
1. H. Bell, Some topological
2. M. Brown, Sets of constant
extensions
distance
of plane geometry
(manuscript).
from a planar
set, Michigan
Math.
J.
19 (1972), 321-3233-
S. Ferry,
4.
R. Gariepy
Minkowski space,
When e-boundaries
and W. D. Pepe,
Proc.
are manifolds,
On the level
Fund.
sets
Math,
(to appear).
of a distance
function
in a
Amer. Math. Soc. 31 (1972), 255—259-
DEPARTMENT OF MATHEMATICS, UNIVERSITY OF TORONTO, TORONTO.ONTARIO,
CANADA
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