@
Appl. Gen. Topol. 23, no. 2 (2022), 453-462

doi:10.4995/agt.2022.16641

© AGT, UPV, 2022

The ε-approximated complete invariance
property

G. Garćıa

Departamento de Matemáticas, UNED, 03202 Elche, Alicante, Spain (gonzalogarciamacias@gmail.com)

Dedicated to my teacher and friend Prof. Dr. Gaspar Mora

Communicated by O. Okunev

Abstract

In the present paper we introduce a generalization of the complete
invariance property (CIP) for metric spaces, which we will call the ε-
approximated complete invariance property (ε-ACIP). For our goals, we
will use the so called degree of nondensifiability (DND) which, roughly
speaking, measures (in the specified sense) the distance from a bounded
metric space to its class of Peano continua. Our main result relates the
ε-ACIP with the DND and, in particular, proves that a densifiable
metric space has the ε-ACIP for each ε > 0. Also, some essentials
differences between the CIP and the ε-ACIP are shown.

2020 MSC: 54H25; 54C50; 54C10 ; 54E40.

Keywords: complete invariance property (CIP); set of fixed points; Peano
continua; α-dense curves; degree of nondensifiability.

1. Introduction

In 1973 Ward [20] introduced the following concept:

Definition 1.1. A topological space X has the complete invariance property
(CIP) if for every non-empty and closed C ⊂ X there is a continuous mapping
f : X −→ X such that Fix(f) = C, where Fix(f) stands for the set of fixed
points of f.

Received 11 November 2021 – Accepted 16 August 2022

http://dx.doi.org/10.4995/agt.2022.16641
https://orcid.org/0000-0001-6840-3226


G. Garćıa

As is mentioned in [8], some spaces known to have the CIP include n-cells,
dendrites, convex subsets of Banach spaces, compact manifolds without bound-
ary, and all compact triangulable manifolds with or without boundary.

It is convenient to recall that a Peano continuum is a compact, connected and
locally connected metric space (X,d), or equivalently, by the Hahn-Mazurkiewicz
Theorem (see, for instance, [19, 21]), X is the continuous image of the unit in-
terval I = [0, 1].

In [20] was asked the following:

Has every Peano continuum the CIP?

The answer is negative: in [8, 9] are given some examples of n-dimensional
Peano continua, with n > 1, that fail to have the CIP. However, for n = 1 the
situation is very different:

Theorem 1.2 (Martin and Tymchatyn [10], 1980). Every 1-dimensional Peano
continuum has the CIP.

Since the publication of the Ward’s paper, many others works have been
devoted to the study and analysis of the CIP and other issues related with it,
see [2, 5, 6, 7, 11, 12, 13, 22] and references therein. So, it seems that the study
of the CIP problem, and its variants, is an interesting and actual topic.

On the other hand, the so called degree of nondensifiability (DND), explained
in detail in Section 2, has been used to prove, under suitable conditions, the
existence of fixed points of continuous self mappings defined into a non-empty,
bounded, closed and convex subset of a Banach space (see [3] and references
therein). In the present paper, for a given metric space (X,d), we introduce
the concept of ε-approximated complete invariance property (ε-ACIP), which
generalizes the CIP one and, by using the DND, we relate in our main result (see
Theorem 3.2) this novel concept with the DND of a bounded metric space. In
particular, our main result proves that densifiable metric spaces (and therefore
every Peano continuum) have the ε-ACIP for each ε > 0.

Also, and as consequence of our main result, we derive some properties
for the ε-ACIP which are not satisfied by the CIP, namely, that the ε-ACIP is
preserved (in the specified sense) by the countable or finite products of bounded
metric spaces or by the continuous image of a bounded metric space.

2. The degree of nondensifiability

In this section, and for a better comprehension of the manuscript, we recall
the concepts of α-dense curves and densifiable sets and also that of degree of
nondensifiability. As in Section 1, (X,d) will be a metric space and we denote
by B(X) the class of non-empty and bounded subsets of X.

In 1997 Cherruault and Mora introduced in [15] the following concepts:

Definition 2.1. Let α ≥ 0 and B ∈ B(X). A continuous mapping γ : I −→
(X,d) is said to be an α-dense curve in B if it satisfies:

(i) γ(I) ⊂ B.
(ii) For any x ∈ B there is y ∈ γ(I) such that d(x,y) ≤ α.

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The ε-approximated complete invariance property

The class of α-dense curves in B is denoted by Γα,B. The set B is said to be
densifiable if Γα,B 6= ∅ for each α > 0.

For a detailed exposition of the α-dense curves and densifiable sets, see
[1, 14, 17]. Some comments are necessary before to continue:

(I) Let us note that, given B ∈ B(X), Γα,B 6= ∅ for each α ≥ Diam(B),
the diameter of B. Indeed, fixed x0 ∈ B, the mapping γ(t) = x0 is an
α-dense curve in B for each α ≥ Diam(B).

(II) If B = In for some integer n > 1 then a 0-dense curve is, precisely,
a space-filling curve (see [19]), i.e. a continuous mapping from I onto
In. So, we can say that the α-dense curves are a generalization of the
space-filling curves.

(III) By recalling that the Hausdorff distance between B1,B2 ∈B(X) is given
by

dH(B1,B2) = max
{

sup
b1∈B1

inf
b2∈B2

d(b1,b2), sup
b2∈B2

inf
b1∈B1

d(b1,b2)
}
,

is clear that if γ is an α-dense curve in B ∈B(X), then dH(B,γ(I)) ≤ α.
We also recall that dH is pseudometric, and is a metric if X is complete,
and a metric in the class of non-empty, bounded and closed subsets of
X.

Next, we show some examples.

Example 2.2 (A compact and connected but not densifiable set). Let, in the
Euclidean plane, the set

B =
{

(x, sin(1/x)) : x ∈ [−1, 0) ∪ (0, 1]
}⋃{

(0,y) : y ∈ [−1, 1]
}
.

Then, given any continuous γ : I −→ R2 with γ(I) ⊂ B, γ(I) has to be
contained is some of the three connected components of B. So, if 0 < α < 1,
there is not an α-dense curve in B, and consequently B is not densifiable.

Example 2.3 (A densifiable set without the CIP). Consider, in the Euclidean
plane, the sets

B1 =
{

(x, sin(1/x)) : x ∈ (0, 1]
}
, B2 =

{
(0,y) : y ∈ [−1, 1]

}
,

and let B = B1 ∪B2, often called the topologist’s sine. Then, is easy to prove
that B is densifiable. In the following lines we will show that B has not the
CIP.

Define the set
C = B ∩ (I × [−1, 0]),

and assume that there is a continuous f : B −→ B such that f(C) = C. As
C ∩B1 = f(C ∩B1) ⊂ f(B1) and f(B1) is path-wise connected, f(B1) = B1.
Hence, as B is compact, we have B = B1 = f(B1) ⊂ f(B) = f(B), where
the bar stands for the closure. This means that f is surjective and therefore
f(B2) = B2.

So, there is a continuous surjection ϕ : [−1, 1] −→ [−1, 1] such that f(0,y) =
(0,ϕ(y)) for all y ∈ [−1, 1]. Hence there exists a ∈ [−1, 1] such that ϕ(a) = 1.

© AGT, UPV, 2022 Appl. Gen. Topol. 23, no. 2 455



G. Garćıa

Set b = ϕ(1). As [−1, 0] is the set of fixed points of ϕ, we conclude that
a ∈ (0, 1) and b ∈ [−1, 1).

Next, define ψ : [a, 1] −→ [−1, 1]2 as ψ(x) = (x,ϕ(x)) and denote by ∆ the
diagonal of [−1, 1]2. Let us note that ψ([a, 1]) ∩ ∆ = ∅ because ϕ does not
have any fixed point in [a, 1]. Hence, ψ is a path (see the below definition) in
[−1, 1] \ ∆.

But, the set [−1, 1]\∆ has two components Ω1 and Ω2 which are above and
below of ∆, respectively. Then, ψ(a) = (a, 1) ∈ Ω1 and ψ(1) = (1,b) ∈ Ω2,
which is contradictory. So, B does not have the CIP as claimed.

Following Willard [21], we recall that a topological space Y is said to be
path-wise connected (resp. arc-wise connected ) if for any x,y ∈ B there is a
continuous (resp. a one-to-one continuous) f : I −→ B, often called path (resp.
arc) such that f(0) = x and f(1) = y. However, if Y is a Hausdorff space (and,
in particular, a metric space), both concepts are equivalents (see [21, Corollary
31.6]. Here, as we work with metric spaces, for our goals is more convenient to
use the term arc-wise connected.

At this point, we can state the following result (see [17]):

Proposition 2.4. Let B ∈B(X) be arc-wise connected. Then B is densifiable
if, and only if, it is precompact.

Although, by the Hahn-Mazurkiewicz Theorem, In is a Peano continuum
and in particular densifiable, the above result also demonstrates us that In is
densifiable. Moreover, we can give an explicit expression of an α-dense curve
in In, γ, for an arbitrarily small α > 0, such that γ(I) is also a 1-dimensional
Peano continuum:

Example 2.5 (1-dimensional Peano continua densifying In). Fixed n > 1, for
a given integer k ≥ 1 define γk : I −→ Rn as

γk(t) =
(
t,

1

2
(1 − cos(πmt)), . . . ,

1

2
(1 − cos(πmk−1t))

)
,

for all t ∈ I. Then, γk is a
√
n−1
k

-dense curve in In (see [1, Proposition 9.5.4])

Remark 2.6. Other examples of α-dense curves in more general subsets of Rn
than In can be found in [18].

From the concepts of α-dense curves, we can define the so called degree of
nondensifiability, which was introduced by Mora and Mira in [16] and analyzed
in [4]:

Definition 2.7. Given B ∈ B(X), we define the degree of nondensifiability,
DND, of B as

φd(B) = inf
{
α ≥ 0 : Γα,B 6= ∅

}
.

As we have pointed out above, Γα,B 6= ∅ for each α ≥ Diam(B) and therefore
the DND is well defined. Also, let us note that, for a given B ∈ B(X), φd(B)

© AGT, UPV, 2022 Appl. Gen. Topol. 23, no. 2 456



The ε-approximated complete invariance property

measures (in the specified sense) the distance from B to the class of its Peano
continua.

Example 2.8 (see [16]). Let B be the closed unit ball of a Banach space V ,
and d the distance in V induced by its norm. Then,

φd(B) =




0, if V is finite dimensional

1, if V is infinite dimensional
.

Some properties of the DND are given in the next result. (see [4, 16]).

Proposition 2.9. The DND satisfies the following:

(1) If φd(B) = 0, then B is precompact. Moreover, if B is precompact and
arc-wise connected then φd(B) = 0.

(2) φd(B) = φd(B), for each B ∈B(X) where, as usual, the bar stands for the
closure.

On the other hand, for our main result we will use Theorem 1.2 and the
DND. So, we will need that the α-dense curves used in the definition of the
DND be 1-dimensional Peano continua. Note that, a priori, an α-dense curve is
not necessarily a 1-dimensional Peano continua: for instance, a n-dimensional
Peano continua or, in particular, the space-filling curves in In given in [19].
However, in the next result, we prove that the DND can be defined by means
of α-curves such that the image of I under these curves be a 1-dimensional
Peano continua.

Theorem 2.10. Given B ∈ B(X) and α > 0, let Γ(1)α,B ⊂ Γα,B be the class of
α-dense curves in B such that γ(1)(I) is a 1-dimensional Peano continuum for

all γ(1) ∈ Γ(1)α,B. By putting

φ
(1)
d (B) = inf

{
α ≥ 0 : Γ(1)α,B 6= ∅

}
,

we have φd(B) = φ
(1)
d (B).

Proof. Let α be such that α > φd(B) and γ : I −→ (X,d) an α-dense curve
in B. So, by the compactness of γ(I), given any ε > 0 there exists a finite set
{y1, . . . ,yn}⊂ γ(I) (without loss of generality we assume n > 1) such that

(2.1) B ⊂
n⋃
i=1

B̄d(yi,α + ε),

B̄d(yi,α + ε) being the closed ball centered at yi of radius α + ε.
As γ(I) is a Peano continuum it is arc-wise connected (see, for instance, [21,

Theorem 31.2]), for each i = 1, . . . ,n− 1 there exists a one-to-one continuous
hi : I −→ γ(I) with hi(0) = yi and hi(1) = yi+1. In particular, each hi(I) is a 1-
dimensional Peano continuum, for i = 1, . . . ,n. Define, for each i = 1, . . . ,n−1,

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G. Garćıa

the one-to-one continuous τi : I −→ [ i−1n−1,
i
n

] as τi(t) =
i−1+t
n−1 for all t ∈ I.

Then, the mapping γ(1) : I −→ (X,d) given by

γ(1)(t) = hi
(
τi(t)

)
, for t ∈ [

i− 1
n− 1

,
i

n
], i = 1, . . . ,n− 1,

is continuous, γ(1)(I) ⊂ γ(I) ⊂ B and γ(1)(I) is a 1-dimensional Peano con-
tinuum because it is the finite union of 1-dimensional Peano continua. Also,

from (2.1) we have γ(1) ∈ Γ(1)α+ε,B . By the arbitrariness of ε > 0, we con-
clude that φ

(1)
d (B) ≤ α and by the arbitrariness of α > φd(B), the inequality

φ
(1)
d (B) ≤ φd(B) holds.

On the other hand, if γ ∈ Γ(1)α,B, from the inclusion Γ
(1)
α,B ⊂ Γα,B, we have

γ ∈ Γα,B. Thus, φd(B) ≤ φ
(1)
d (B) and the proof is now complete.

�

To conclude this section, we give a result for the DND of the product of
bounded metric spaces.

Proposition 2.11. Let Λ be a finite set or Λ = N, and (Xλ,dλ)λ∈Λ a family of
metric spaces such that Diam(Xλ) ≤ M for certain M > 0 and all λ ∈ Λ. Put
φ∗ = sup{φdλ(Xλ) : λ ∈ Λ}, X

∗ =
∏
λ∈Λ Xλ and d

∗(x,y) = max{dλ(x,y) :
λ ∈ Λ} if Λ is finite or d∗(x,y) =

∑
k≥1 2

−kdk(x,y) if Λ = N, for all x,y ∈ X∗.
Then,

φd∗(X
∗) ≤ φ∗.

Moreover if Λ is finite, then the equality holds.

Proof. Firstly, note that (X∗,d∗) is, effectively, a bounded metric space and
therefore φd∗(X

∗) is well defined (in fact, φd∗(X
∗) ≤ M).

Assume, Λ = N and let α > φ∗. Let, for each k ≥ 1, γk : I −→ Xk an
α-dense curve in Xk. So, for each k ≥ 1, given xk ∈ Xk there is tk ∈ I such
that

(2.2) dk(xk,γk(tk)) ≤ α.

Let ω = (ωk)k≥1 : I −→ IN be a space-filling curve (see [19, Section 7.5]).
That is, ω (and hence each coordinate function ωk) is continuous and ω(I) = I

N.
Define γ : I −→ X∗ as

γ(t) =
(
γk(ωk(t))

)
k≥1, for all t ∈ I.

It is clear that γ is continuous and γ(I) ⊂ X∗. Also, given (xk)k≥1 ∈ X∗
take (tk)k≥1 ⊂ I satisfying (2.2) and t ∈ I such that ω(t) = (tk)k≥1. So, we
have

d∗
(
(xk)k≥1,γ(t)

)
=
∑
k≥1

dk
(
xk,γk(ωk(t))

)
2k

=
∑
k≥1

dk
(
xk,γk(tk)

)
2k

≤ α.

and consequently γ is an α-dense curve in X∗. Then, φd∗(X
∗) ≤ α and letting

α → φ∗, we conclude φd∗(X∗) ≤ φ∗.

© AGT, UPV, 2022 Appl. Gen. Topol. 23, no. 2 458



The ε-approximated complete invariance property

If Λ is finite, without loss of generality we assume Λ = {1, . . . ,n} for some
n > 1, we take ω = (ω1, . . . ,ωn) : I −→ In a space-filling curve (again, [19])
and the proof follows in a totally analogous way that above.

Assume φd∗(X
∗) < φ∗ and take φd∗(X

∗) < α < φ∗ and an α-dense curve in
X∗, put γ = (γ1, . . . ,γn) : I −→ (X∗,d∗). Then, fixed 1 ≤ k ≤ n, the mapping
γk : I −→ (Xk,dk) is continuous and one can check straightforwardly that it is
an α-dense curve in Xk. But, this is not possible as α < φ

∗ ≤ φdk(Xk).
�

3. The main result

We start this section with the following definition:

Definition 3.1. Given ε ≥ 0, we will say that a metric space (X,d) has the ε-
approximated complete invariance property (ε-ACIP) if for each non-empty and
closed C ⊂ X there is a continuous fε : X −→ X such that dH(C, Fix(fε)) ≤ ε.

The following facts are clear from the definitions:

(I) If (X,d) is bounded, then (X,d) has ε-ACIP for every ε ≥ Diam(X).
(II) The 0-ACIP is, precisely, the CIP. Also, the CIP implies the ε-ACIP for

each ε > 0, but as we will see below, the inverse implication does not
hold in general. That is to say, there are metric spaces with the ε-ACIP
for all ε > 0, but such metric spaces do not have the CIP.

Now, we are ready to state and prove our main result:

Theorem 3.2. Let (X,d) a bounded metric space. Then, (X,d) has the ε-
ACIP for each ε > φd(X). In particular, if X is densifiable then it has the
ε-ACIP for each ε > 0.

Proof. Let ε be such that ε > φ(X). Let any C ⊂ X non-empty and closed,
and γε : I −→ (X,d) and ε-dense curve such that γε(I) is a 1-dimensional
Peano continuum. Such ε-dense curve exists by virtue of Theorem 2.10.

Define the set

GC =
{
x ∈ γε(I) : d(x,c) ≤ ε, for some c ∈ C

}
⊂ X.

It is clear that the set GC is non-empty and closed. Thus, by Theorem 1.2,
there is fε : X −→ X with Fix(fε) = GC.

Now, let c ∈ C. As γε is an ε-dense curve in X, there is x ∈ γε(I)
with d(x,c) ≤ ε. Then, x ∈ GC and therefore x = fε(x). So, we have
infx∈Fix(fε) d(c,x) ≤ ε and from the arbitrariness of c ∈ C, we infer
(3.1) sup

c∈C
inf

x∈Fix(fε)
d(c,x) ≤ ε.

Likewise for a given x ∈ Fix(fε), as x ∈ GC, d(c,x) ≤ ε for some c ∈ C.
Consequently, infc∈C d(c,x) ≤ ε and noticing the arbitrariness of x ∈ Fix(fε)
(3.2) sup

x∈Fix(fε)
inf
c∈C

d(c,x) ≤ ε.

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G. Garćıa

So, from (3.1) and (3.2), we have dH(C, Fix(fε)) ≤ ε.
If X is densifiable then, by the definition of the DND, φd(X) = 0 and

therefore has the ε-ACIP for each ε > 0.
�

An immediate consequence of the above result is the following:

Corollary 3.3. Every Peano continuum has the ε-ACIP for each ε > 0.

As we have said above, in general, the ε-ACIP for each ε > 0 does not imply
the CIP. We illustrate this fact in the following examples.

Example 3.4. Let X be the topologist’s sine of Example 2.3. Then, X is
densifiable but does not have the CIP. However, by Theorem 3.2 X has the
ε-ACIP for each ε > 0.

Example 3.5. Let X be a n-dimensional Peano continuum without the CIP
(see [8, 9]). Then, by Corollary 3.3, X has the ε-ACIP for each ε > 0.

So, in general, we cannot replace the condition ε > φd(X) by ε ≥ φd(X) in
Theorem 3.2. This fact is explained by the following ones:

(I) There is not necessarily a φd(X)-dense curve in X. Indeed, for instance,
the topologist’s sine X of Example 2.3 satisfies φd(X) = 0 but there is
not a 0-dense curve in X: otherwise, X would be a Peano continuum,
which is not possible because it is not locally connected.

(II) Even if X = γ(I), for certain continuous γ : I −→ X, if γ(I) is not a
1-dimensional Peano continuum, we cannot apply Theorem 1.2 in the
proof of Theorem 3.2 to derive that X has the CIP (see also Example
3.5).

We have remarked above that if (X,d) is bounded, then (X,d) has ε-ACIP
for every ε ≥ Diam(X). This bound can be improved by Theorem 3.2:

Example 3.6. Let X be the set given in Example 2.2. Then, Diam(X) = 2
and φd(X) = 1. So, by Theorem 3.2, X has ε-ACIP for every ε > 1.

As was proved in [6], the CIP need not be preserved by self-products. How-
ever, bearing in mind Proposition 2.11 and Theorem 3.2, we have the following
result for the product of bounded metric spaces:

Corollary 3.7. With the notation of Proposition 2.11, (X∗,d∗) has the ε-
ACIP for each ε > φ∗. In particular, the finite or countable product of Peano
continua has the ε-ACIP for each ε > 0.

Example 3.8. Let (X,d) be the 1-dimensional Peano continuum given in [6,
Theorem 2.2]. Then, X×X does not have the CIP. However, by Corollary 3.7,
X ×X has the ε-ACIP for each ε > 0.

Also, the CIP need not to be preserved by continuous mappings. Indeed, take
any metric space (X,d) that does not have the CIP and τ the discrete topology
on X. Then, (X,τ) has the CIP and the identity mapping g : (X,τ) −→ (X,d)
is continuous. However, for the ε-ACIP we have the following:

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The ε-approximated complete invariance property

Corollary 3.9. Let (X,d) and (Y,d′) be bounded metric spaces and g : (X,d) −→
(Y,d′) continuous. Then (Y,d′) has the ε-ACIP for each ε > ωφd(X)(g), where

ωr(g) = sup
{
d′(f(x),f(y)) : x,y ∈ X,d(x,y) ≤ r

}
,

is the modulus of continuity of g of order r, for r ≥ 0.

Proof. It is immediate to check that if γ : I −→ (X,d) is an α-dense curve in
(X,d), then g◦γ : I −→ (Y,d′) is a ωα(g)-dense curve in (Y,d′). Therefore, we
infer that φd′(Y ) ≤ ωφd(X)(g) and by Theorem 3.2, (Y,d

′) has the ε-ACIP for
each ε > ωφd(X)(g).

�

On the other hand, it is important to stress that the reciprocal of Theorem
3.2 is not true in general: there are metric spaces with the ε-ACIP for all ε > 0
(in fact, with the CIP) that are not densifiable:

Example 3.10. Let X be the closed unit ball of an infinite dimensional Banach
space. From the comments of Section 1, X has the CIP and therefore the ε-
ACIP for all ε > 0. However, from Example 2.8, φd(X) = 1 and noticing
Proposition 2.9 X is not densifiable.

So, we conclude our exposition with the following question:

If (X,d) is a bounded metric space having the ε-ACIP, for
some ε > 0, under what conditions can we relate, in some

way, ε and φd(X)?

Acknowledgements. The author is very grateful to the anonymous referee
for his/her comments and suggestions, which have substantially improved the
presentation of the paper.

References

[1] Y. Cherruault and G. Mora, Optimisation Globale. Théorie des Courbes α-denses,
Económica, Paris, 2005.

[2] R. Dubey and A. Vyas, Wavelets and the complete invariance property, Mat. Vesnik, 62

(2010), 183–188.
[3] G. Garćıa and G. Mora, A fixed point result in Banach algebras based on the degree of

nondensifiability and applications to quadratic integral equations, J. Math. Anal. Appl.

472 (2019), 1220–1235.
[4] G. Garćıa and G. Mora, The degree of convex nondensifiability in Banach spaces, J.

Convex Anal. 22 (2015), 871–888.

[5] K. H. Heinrich and J. R. Martin, G-spaces and fixed point sets, Geom. Dedicata 83
(2000), 39–61.

[6] J. R. Martin, Fixed point sets of metric and nonmetric spaces, Trans. Amer. Math. Soc.

284 (1984), 337–353.

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G. Garćıa

[7] J. R. Martin, Fixed point sets of LC∞, C∞ continua, Proc. Amer. Math. Soc. 81 (1981),
325–328.

[8] J. R. Martin, Fixed point sets of Peano continua, Pacific J. Math. 74 (1978), 163–166.
[9] J. R. Martin and S. B. Nadler, Examples and questions in the theory of fixed point sets,

Canad. J. Math. 31 (1979), 1017–1032.

[10] J. R. Martin and E. D. Tymchatyn, Fixed point sets of 1-dimensional Peano Continua,
Pacific J. Math. 89 (1980), 147–149.

[11] D. Masood and P. Singh, Complete invariance property on hyperspaces, JP J. Geom.

Topol. 17 (2015), 83–94.
[12] D. Masood and P. Singh, On equivariant complete invariance property, Sci. Math. Jpn.

77 (2013), 1–6.

[13] S. C. Maury, Hyperspaces and the S-equivariant complete invariance property, Kyung-
pook Math. J. 55 (2015), 219–224.

[14] G. Mora, The Peano curves as limit of α-dense curves, Rev. R. Acad. Cienc. Exactas
F́ıs. Nat. Ser. A Math. RACSAM 9 (2005), 23–28.

[15] G. Mora and Y. Cherruault, Characterization and generation of α-dense curves, Com-

puters Math. Applic. 33 (1997), 83–91.
[16] G. Mora and J. A. Mira, Alpha-dense curves in infinite dimensional spaces, Int. J. Pure

Appl. Math. 5 (2003), 437–449.

[17] G. Mora and D. A. Redtwitz, Densifiable metric spaces, Rev. R. Acad. Cienc. Exactas
F́ıs. Nat. Ser. A Math. RACSAM 105 (2011), 71–83.

[18] M. Rahal, R. Ziadi and A. Ellaia, Generating α-dense curves in non-convex sets to solve

a class of non-smooth constrained global optimization, Croatian Operational Research
Review 10 (2019), 289–314.

[19] H. Sagan, Space-Filling Curves, Springer-Verlag, New York 1994.

[20] L. E. Ward, Fixed point sets, Pacific J. Math. 47 (1973), 553–565.
[21] S. Willard, General Topology, Dover Pub. Inc., New York 1970.

[22] D. X. Zhou, Complete invariance property with respect to homeomorphism over frame
multiwavelet and super-wavelet spaces, Journal of Mathematics 2014 (2014), Article ID

528342, 6 pages.

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