A comparison of two notions of porosity
Filip Strobin
Institute of Mathematics
Polish Academy of Sciences
Śniadeckich 8, P.O. Box 21
00-956, Warszawa, Poland
email: [email protected]
Abstract
In the paper we compare two notions of porosity: the R-ball porosity (R > 0)
de…ned by Preiss and Zajíµcek, and the porosity which was introduced by Olevskii
(here it will be called the O-porosity). We …nd this comparison interesting since in
the literature there are two similar results concerning these two notions. We restrict
our discussion to normed linear spaces since the R-ball porosity was originally de…ned
in such spaces.
1
Introduction
Olevskii [2] proved that every convex nowhere dense subset of a Banach space is O-porous.
On the other hand, Zajíµcek [6] in his survey paper on porosity recalled the de…nition of
R-ball porosity (R > 0) and observed (without giving a proof) that a convex nowhere
dense subset of a Banach space is R-ball porous for every R > 0 (cf. [6, p. 518]). Zajíµcek
also considered the de…nition of the ball smallness (a set M is ball small if M = [n2N Mn
and each Mn is Rn -ball porous for some Rn > 0).
In the paper we prove that every R-ball porous subset of any normed linear space is
O-porous; moreover, with the help of the Baire Category Theorem, we show that in every
nontrivial Banach space there exists an O-porous set which is not ball small (hence not
R-ball porous for any R > 0).
We also show that, in general, the notion of O-porosity is more restrictive than the
most "natural" notions of porosity ((c-)lower or (c-)upper porosity; see [6]).
Also, we would like to point out that Olevskii, as an application of his result, gave a
few natural examples of sets which are countable unions of convex nowhere dense subsets
of Banach spaces (one of them deals with the Banach-Steinhaus Theorem), so Zajíµcek’s
observation implies that these sets are ball small. This seems to be interesting since
Zajíµcek [6, p. 516] stated that there was no result in the literature which asserted that an
"interesting" set of singular points is ball small.
This paper is organized as follows. First, in the next section, we give exact de…nitions
of some types of porosity and discuss basic relationships between them. Then, in Section
3, we prove the main result of this note.
2
Some notions of porosity
We start with presenting de…nitions of ( -)O-porosity, R-ball porosity, the ball smallness
and c-lower porosity (c > 0).
1
Let (X; k k) be a normed linear space and M X. Given x 2 X and r > 0, we denote
by B(x; r) the open ball with center x and radius r.
De…nition 1 ([2]) M is called O-porous if
8
2(0;1)
9R0 >0 8x2M 8R2(0;R0 ) 9y2X (kx
yk = R and B(y; R) \ M = ?) :
Remark 1 In fact, De…nition 1 was suggested to Olevskii by K. Saxe (see [2, Remark 1]).
De…nition 2 ([6]) Let R > 0. We say that M is R-ball porous if
8x2M 8
2(0;1)
9y2X (kx
yk = R and B(y; R) \ M = ?) :
Remark 2 The de…nition of R-ball porosity presented in [6] ([6, p. 516]) is slightly
di¤erent from the above one. Namely, M is R-ball porous if
8x2M 8"2(0;R) 9y2X (kx
yk = R and B(y; R
") \ M = ?) :
However, it is easy to see that they are equivalent.
De…nition 3 ([6]) Let c > 0. M is called c-lower porous if for any x 2 M; we have that
2 lim inf
R!0+
where (x; R; M ) := supfr
(x; R; M )
R
0 : 9y2X B(y; r)
c;
B(x; R)nM g:
De…nition 4 ([6]) We say that M is lower porous if for any x 2 M , we have that
lim inf
R!0+
(x; R; M )
> 0:
R
If we substitute the upper limit for the lower limit in the above de…nitions, we get
de…nitions of c-upper porosity and upper porosity.
Remark 3 Clearly, the following implications hold:
1 c-lower porosity =) lower porosity =) upper porosity;
2 c-lower porosity =) c-upper porosity =) upper porosity.
De…nition 5 ([6]) M is called ball small if there exist a sequence (Mn )n2N of sets and
a sequence (Rn )n2N of positive reals such that M = [n2N Mn and each Mn is Rn -ball
porous.
De…nition 6 ([2]) M is called -O-porous if M is a countable union of O-porous sets.
It is obvious that the above de…nitions can be formulated in any metric space. However,
this paper deals only with normed linear spaces and from now on we assume that the letter
X denotes a normed linear space (some results presented below may be false for metric
spaces).
We omit an obvious proof of the following
2
Proposition 1 If x 2 X and M
X is c-lower porous [O-porous,
porous, ball small], so is its translation M + x:
Proposition 2 Let M
-O-porous, R-ball
X and c > 0: The following conditions are equivalent:
(i) M is c-lower porous;
(ii) 8x2M 8
2(0; 21 c)
9R0 >0 8R2(0;R0 ) 9y2X B(y, R)
Proof. (i) =) (ii)
Take any x 2 M ,
2 (0; 2c ) and
B(x; R)nM:
2 ( ; 2c ). By (i), there exists R0 > 0 such that
inf
R2(0;R0 )
(x; R; M )
R
:
(1)
Now let R 2 (0; R0 ). By (1), (x; R; M )
R. Hence and by the fact that R > R, we
have that there exists y 2 X such that B(y; R) B(x; R)nM .
(ii) =) (i)
Fix any x 2 M and 2 (0; 2c ). Take R0 as in (ii). It su¢ ces to show that
inf
R2(0;R0 )
(x; R; M )
R
:
Take any R 2 (0; R0 ). By (ii), there exists z 2 X such that B(z; R)
B(x; R)nM .
Hence R 2 fr 0 : 9y2X B(y; r) B(x; R)nM )g, so (x; R; M )
R and, in particular,
(x;R;M )
.
R
Proposition 3 If ? 6= M
X is c-lower porous, then c
1.
Proof. Since X is a normed space, the following clearly holds:
8d> 1 8R>0 8x2X 8z2X (x 2
= B(z; dR) =) B(z; dR) " B(x; R)) :
2
Hence we infer 2 (x; R; M )
R for any x 2 M and R > 0:
Proposition 4 Every O-porous set M
X is 1-lower porous.
Proof. Let M be O-porous. By Proposition 2, it su¢ ces to prove
8x2M 8
2(0; 12 )
9R0 >0 8R2(0;R0 ) 9y2X B(y; R)
B(x; R)nM:
Fix x 2 M and 2 (0; 21 ). Take R0 > 0 as in the de…nition of O-porosity, chosen for
:= 2 . Take any R < 2R0 . Since 21 R < R0 ; there exists y 2 X such that k y x k= 21 R
and B(y; 12 R) \ M = ?. Since B(y; 12 R) = B(y; R), it is enough to show that
B(y; R)
B(x; R):
For any z 2 B(y; R); we have
kx
zk
kx
yk+ky
zk
(2)
1
R + R < R;
2
so (2) holds.
Now we give an example of 1-lower porous subset of R which is not O-porous. This example
and Proposition 4 show that, in general, the notion of O-porosity is more restrictive than
the notion of 1-lower porosity.
3
Example 1 Consider the set M := [n2N fn +
M is 1-lower porous and is not O-porous.
k
n
: k = 0; :::; n
1g: It is easy to see that
By Proposition 4, every -O-porous set is -1-lower porous (a countable union of
1-lower porous sets). However, we do not know if there exists a set which is -1-lower
porous and is not -O-porous. We leave it as an open question.
3
Main results
We …rst show that any R-ball porous subset of a normed linear space is O-porous. We
need the following result which is also given (without a proof) in [1].
Proposition 5 Let X be a normed linear space. If M
r-ball porous for all r 2 (0; R]:
X is R-ball porous, then M is
Proof. Fix r 2 (0; R], x 2 M and 2 (0; 1): De…ne := 1 Rr (1
): By hypothesis,
there exists y 2 X such that kx yk = R and B(y; R) \ M = ?: Let y1 2 [x; y] be such
that k y1 x k= r: It su¢ ces to prove that
B(y1 ; r)
B(y; R):
(3)
For any z 2 B(y1 ; r); we have
ky
zk ky
y1 k + k y 1
z k< R
r+ r= R
which yields (3).
Corollary 1 Any R-ball porous subset of a normed linear space is O-porous.
Now we will show that in any nontrivial Banach space there exists an O-porous set
which is not ball small. In particular, the class of -O-porous sets in Banach spaces is
essentially wider (with respect to the inclusion) than the class of ball small sets. Hence
we infer Olevskii’s result does not imply that a countable union of nowhere dense convex
subsets of any Banach space is ball small.
We start with some auxiliary results.
Proposition 6 Let R > 0 and M
R:
1 If M is R-ball porous, then for any x 2 R; the set [x
most 2 elements.
2 If for any x 2 R; the set [x
1
2 R-ball porous.
R; x + R] \ M contains at
R; x + R] \ M contains at most 2 elements, then M is
Proof. Ad 1) Assume that for some x 2 R; the set [x R; x + R] \ M contains more than
two elements. Let a; b; c 2 R be such that a < b < c and fa; b; cg [x R; x + R] \ M:
Therefore c b < 2R and b a < 2R. Take 2 (0; 1) such that
> max
jc
b Rj jb
;
R
a Rj
R
:
It is easy to see that a 2 B(b R; R) and c 2 B(b + R; R). Hence M is not R-ball
porous.
Ad 2) Fix any x 2 M and 2 (0; 1). Since [x R; x + R] \ M contains at most two
elements, we see that (x; x + R] \ M = ? or [x R; x) \ M = ?: Therefore
B x 21 R; 12 R \ M = ? or B x + 12 R; 12 R \ M = ?: Hence M is 12 R-ball porous.
4
Corollary 2 Any R-ball porous subset of R is countable.
Hence, as an immediate consequence, we get the following characterization of ball small
subsets of R. Note that the folowing result was also given (without a proof) in [3].
Corollary 3 Let M
R: M is ball small i¤ M is countable.
Now we will give an example of an O-porous subset of R which is not ball small.
Example 2 Fix a sequence = ( n )n2N[f0g such that n > 21 and n % 1. Consider
the symmetric perfect set C( ) (cf. [5, p. 318]): The set C( ) is de…ned similarly to the
Cantor ternary set: by induction, we de…ne the sequence of sets fCs gs2f0;1g<N satisfying
the following conditions (f0; 1g<N is the set of all …nite sequences of ones and zeros, j s j
means the length of s; i.e.; j s j=j (s(0); :::; s(n 1)) j= n, and diamA denotes the diameter
of A):
(c1) C? : =[0,1];
(c2) if s 2 f0; 1g<N and Hs is the concentric open subinterval of Cs of length jsj ljsj ;
where ljsj =diamCs ; then Cs^0 is the left and Cs^1 is the right subinterval of Cs nHs .
Finally, set C( ) := \n2N [jsj=n Cs : Since C( ) is uncountable, it is not ball small in
view of Corollary 3. We will show that C( ) is O-porous. It is easy to see that for any
1 n 1
n 2 N; ln =
ln 1 . Fix 2 (0; 1) and let n0 2 N be such that
2
n0
+1
:
2
>
(4)
Set R0 := 12 ln0 +1 and let R 2 (0; R0 ). Since ln & 0; there is n 2 N such that
1
ln+1 < R
2
1
ln :
2
(5)
Since R < R0 ; we have that n n0 + 1: Now take any x 2 C( ). In particular, x 2 Cs for
some s with j s j= n. Consider four cases:
Case 1: x 2 Cs^0^1 : Set y := x + R. We will prove that B(y; R) \ C( ) = ?. By (4) and
(5), we have
+1
dist(Cs^0^1 ; Cs^1^0 ) = n ln >
ln ( + 1)R;
2
so it su¢ ces to show that for any z 2 Cs^0^1 ; j z y j
R:
Since diamCs^0^1 = ln+2 < R; for any z 2 Cs^0^1 ; by (4) and (5), we have
jz yj
2R
ln =
1
n 1
2
ln
1
<
1
n+1
ln+1 > R (1 n+1 )R = n+1 R > R:
2
R. Using a similar argument as in Case 1, we may infer
:
R: By (4) and (5), we have
R diamCs^0^1 = R ln+2 = R
Case 2: x 2 Cs^1^0 : Set y := x
B(y; R) \ C( ) = ?.
Case 3: x 2 Cs^0^0 : Set y := x
.
n 1 ln 1
= minfdist(Cq ; Cp ) : p 6= q and j p j=j q j= ng;
and we can show that B(y; R)\C( ) = ? in a similar way as in Case 1.
Case 4: x 2 Cs^1^1 : Set y := x + R: Repeating an argument from Case 3, we infer
B(y; R) \ C( ) = ?.
As a conclusion we see that C( ) is O-porous.
5
.
Lemma 1 Any nontrivial real Banach space (Y; k k0 ) is isometrically isomorphic to some
space (R X; k k); where X is a closed linear subspace of Y and the norm k k satis…es
the following conditions:
(a) k k is equivalent to the norm max k kmax , where
k(t; x)kmax := maxfj t j; kxk0 g for (t; x) 2 R X;
:
(b) 8t2R k(t; 0)k =j t j and 8x2X k(0; x)k = kxk0 .
Proof. Take any x0 2 Y with kx0 k0 = 1 and consider one dimensional subspace
Y1 := ftx0 : t 2 Rg
Y . Since dim Y1 < 1, there exists a closed subspace Y2
Y
such that Y = Y1 Y2 (see, e.g., [4]): Set X = Y2 . Note that for every y 2 Y there are
t 2 R and x 2 X such that y = tx0 + x. For (t; x) 2 R X de…ne k(t; x)k := ktx0 + xk0 .
Observe that k k is a norm on R X, equivalent to the norm k kmax : Then the function
(t; x) 7! tx0 + x is an isometrical isomorphism between (R X; k k) and (Y; k k0 ):
Remark 4 Let (X; k k) be a complex space. Then we can consider it as a real space (it
is obvious how to formalize this statement). It is clear that for any M X; the following
statements are equivalent:
1. M is O-porous [ball small] in (X; k k);
2. M is O-porous [ball small] in (X; k k) considered as a real space.
From now on the symbol (R X; k k) denotes a real normed linear space R X; where
X is a real normed linear space and the norm k k satis…es conditions (a) and (b) from
Lemma 1.
Remark 5 By the above Remark and Lemma 1, in order to prove that any nontrivial
Banach space admits an O-porous set which is not ball small, it su¢ ces to prove it for
every space (R X; k k), where X is a real Banach space.
Lemma 2 Let (X; k k0 ) be a real normed linear space. For any R > 0 and any " > 0;
there exists > 0 such that for any p 2 X and any t1 ; t0 ; t2 2 R, if t1 < t0 < t2 ,
j t1 t0 j< and j t2 t0 j< ; then the set N := ft1 ; t0 ; t2 g B(p; ") is not R-ball porous
in (R X; k k).
Proof. There exists
2 (0; 1] such that for any (s; x) 2 R
maxfj s j; kxk0 g
Fix R > 0 and " > 0. Let
R "
;
4 4
" := min
We are ready to de…ne
n"
:= " min
R
In particular,
"
k(s; x)k:
(6)
:
(7)
o
;1 :
(8)
R
< R:
4
6
X;
Take any p 2 X and any t1 ; t0 ; t2 2 R such that t1 < t0 < t2 , j t1 t0 j< and j t2 t0 j< .
By Proposition 1, we may assume that (t0 ; p) = (0; 0). It su¢ ces to show that there is
2 (0; 1) such that for any (s; y) 2 R X;
k(s; y)k = R =) B((s; y); R) \ N 6= ?:
Let
2 (0; 1) be such that
R
> max
R
" R
;
R
j t1 j R
;
R
j t2 j
:
(9)
Fix any (s; y) such that k(s; y)k = R. Now we will consider three cases.
.
Case 1: y = 0. Since j s j= R > ; we see that s > t2 > 0 or s < t1 < 0. Assume that
s > t2 > 0. Then (t2 ; 0) 2 N and, by (9), we have
k(s; 0)
(t2 ; 0)k = s
t2 = R
t2
R
t2 < R:
If s < t1 < 0, we proceed similarly.
Case 2: j s j< " and y 6= 0. Since
k(s; 0)k + k(0; y)k < " + k(0; y)k
R = k(s; y)k = k(s; 0) + (0; y)k
and
k(0; y)k = k(s; y)
k(s; y)k + k(s; 0)k < R + ";
(s; 0)k
we get, by (7), that
0 < 3"
R
" < k(0; y)k < R + ":
(10)
Now take z 2 [0; y] such that kzk0 = 3". By (7), (9) and (10), we have (0; z) 2 N and also
k(s; y)
(0; z)k = k(s; 0) + (0; y)
= k(s; 0)k + k(0; y)k
Case 3: j s j
(0; z)k
k(s; 0)k + k(0; y)
(0; z)k
kzk0 = R + 2"
3" < R:
k(0; z)k < " + (R + ")
" and y 6= 0. We may assume, without loss of generality, that s
:=
By (8), =" = min
"
R
In particular, k yk0 < " and
using (9) also, we infer
k(s; y)
t2
:
s
; 1 and by (6), maxfj s j; kyk0 g
t2
R
<
s
min
n"
R
". Set
;1
o
min
1
R. Hence
"
;1 :
kyk0
2 (0; 1), so ( s; y)(= (t2 ; y)) is an element of N; and,
( s; y)k =j 1
j R = (1
)R
1
t2
R
R < R:
With the case s
" we proceed analogously.
Thus we have shown that in each case B((s; y); R) \ N 6= ?:
Lemma 3 Let X be a real normed linear space and M
O-porous in the space (R X; k k).
7
R be O-porous. Then M
.
X is
Proof. For any r > 0; we put
Ar := sup fj t j: (t; x) 2 R
X, k(t; x)k = rg :
(11)
Clearly, in the above de…nition it is enough to consider t
0 or t
0. Since k k is
equivalent to the norm max, we infer 0 < Ar < 1 for any r > 0. It is also easy to prove
that for any r > 0, we have:
rA1 = Ar ;
(12)
By (12), we see that for any (t; x) 2 R
jtj
X;
A1 k(t; x)k:
Now we are ready to prove the Lemma. Fix any
O-porous, there exists R0 > 0 such that
8t2M 8R2(0;R0 ) 9a2R (j a
(13)
2 (0; 1) and
2 ( ; 1). Since M is
t j= R and B(a; R) \ M = ?) :
De…ne R1 := R0 =A1 ; and take any R 2 (0; R1 ) and (t; x) 2 M
(14), there exists a 2 R such that j a t j= RA1 and
(14)
X. Since RA1 < R0 , by
B(a; RA1 ) \ M = ?:
(15)
Assume that a = t + RA1 (we proceed analogously with the case a = t RA1 ) and put
" := (
)RA1 . We will show that there exists (s; y) 2 R X such that
k(t; x)
(s; y)k = R and j a
s j< ":
(16)
By (12) and the statement after (11); there exists (q; z) 2 R X such that q
0,
k(q; z)k = R and RA1 " < q
RA1 . Now de…ne s := t + q and y := x + z. Since
(s; y) (t; x) = (q; z) and j a s j=j RA1 q j; we get (16). It su¢ ces to observe that
B((s; y); R) \ (M
X) = ?:
For any (p; u) 2 B((s; y); R), by (13) and (16), we have
jp
aj jp
sj+js
so, by (15), p 2
= M . Hence (p; u) 2
=M
a j< RA1 + " = RA1 ;
X:
Proposition 7 Let X be a normed linear space and R > 0. If M
so is its closure M .
X is R-ball porous,
Proof. Let M
X be R-ball porous. Fix 2 (0; 1) and x 2 M : Take any 2 ( ; 1)
and choose x1 2 M such that k x x1 k< (
)R. By hypothesis, we infer there exists
y1 2 X such that k y1 x1 k= R and B(y1 ; R) \ M = ?: Put y := y1 + (x x1 ). Then
we have k y x k= R and for any z 2 B(y; R); we see that
k y1
z k k y1
yk+ky
so z 2 B(y1 ; R). Hence B(y; R)
z k< (
)R + R = R;
XnM: Finally, we have
B(y; R) = IntB(y; R)
Int(XnM ) = XnM :
.
Now we are ready to give a more general construction. A very important tool in the proof
of the following theorem is the Baire Category Theorem (note that the idea of using
the Baire Category Theorem to show that some sets are not -porous in some senses, is
commonly known - see, e. g. [6, p. 513]).
8
Theorem 1 In any nontrivial Banach space there exists an O-porous set which is not ball
small.
Proof. We will use Remark 5. Let X be any real Banach space, and let C( )
R be
the set de…ned in Example 2. By Lemma 3, C( ) X is O-porous in (R X; k k). We
will show that C( ) X is not ball small. Assume that C( ) X = [n2N Mn : In view of
Proposition 7, it is enough to show that there exists n0 2 N such that Mn0 is not R-ball
porous for any R > 0. Since C( ) X is a closed subset of a Banach space R X, it
is complete. By the Baire Category Theorem, there exists n0 2 N such that Mn0 is not
nowhere dense in C( ) X. Hence there exist a set K 6= ? open in C( ); p 2 X and
" > 0 such that
C( ) X
K B(p; ") Mn0
= Mn0 :
(17)
Now we show that for any R > 0; the set Mn0 is not R-ball porous. Fix R > 0. It is well
known that the family fCs \ C( ) : s 2 f0; 1g<N g forms a topological base of C( ): Hence
there exists s 2 f0; 1g<N such that Cs \ C( ) K and diamCs < , where > 0 is chosen
for R and " as in Lemma 2. Since Cs \ C( ) is in…nite, the set (Cs \ C( )) B(p; ")
contains some set of the form ft1 ; t0 ; t2 g B(p; ") which, according to Lemma 2; is not
R-ball porous. Hence and by (17), we infer Mn0 is not R-ball porous.
Acknowledgement 1 I am grateful to the referee for many useful suggestions. Also, I
would like to thank Professor Jacek Jachymski for some valuable discussions and remarks,
and Dr. Szymon G÷
ab
¾ for his contribution to the proof of Theorem 1.
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[4] W. Rudin, Functional Analysis, McGraw-Hill, Inc., New York, 1991.
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-porosity, Real Anal. Exchange 13 (1987/1988),
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9