A NOTE ON CONNECTEDNESS IM KLEINEN IN C(X)

Journal of the Korean Society of Mathematical Education Series B The Pure and Applied Mathematics.
2015.
May,
22(2):
139-144

- Received : November 18, 2014
- Accepted : February 12, 2015
- Published : May 31, 2015

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Abstract. In this paper, we investigate the relationships between the space
X
and the hyperspace
C
(
X
) concerning admissibility and connectedness im kleinen. The following results are obtained: Let
X
be a Hausdorff continuum, and let
A
∈
C
(
X
). (1) If for each open set
U
containing
A
there is a continuum
K
and a neighborhood
V
of a point of
A
such that
V
⊂
IntK
⊂
K
⊂
U
, then
C
(
X
) is
connected im kleinen
. at
A
. (2) If
IntA
≠ ø, then for each open set
U
containing
A
there is a continuum
K
and a neighborhood
V
of a point of
A
such that
V
⊂
IntK
⊂
K
⊂
U
. (3) If
X
is
connected im kleinen
. at
A
, then
A
is admissible. (4) If
A
is admissible, then for any open subset
U
of
C
(
X
) containing
A
, there is an open subset
V
of
X
such that
A
⊂
V
⊂ ∪
U
. (5) If for any open subset
U
of
C
(
X
) containing
A
, there is a subcontinuum
K
of
X
such that
A
∈
IntK
⊂
K
⊂
U
and there is an open subset
V
of
X
such that
A
⊂
V
⊂ ∪
IntK
, then
A
is admissible.
X
be a Hausdorff continuum, and let 2
^{X}
(
C
(
X
),
K
(
X
),
C_{K}
(
X
)) the hyperspace of nonempty closed subsets(connected closed subsets, compact subsets, continua) of
X
with the Vietoris topology. Throughout by a
continuum
we mean a compact connected Hausdorff space. For a continuum
X
,
C
(
X
) is endowed with the Vietoris topology and, since
X
is a continuum, the hyperspace
C
(
X
) is also a continuum
[8]
.
Wojdyslawsk
[13]
established the conditions of local connectedness between a space
X
and its hyperspace 2
^{X}
(
C
(
X
)). Goodykoontz
[3
,
4
,
5]
investigated local connectedness as a pointwise property in the hyperspace 2
^{X}
(
C
(
X
)) of metric continua. And Goodykoontz and Rhee
[6]
investigated the relationships between the space
X
and the hyperspaces concerning the properties of local compactness and local connectedness. They proved that a Hausdorff space
X
is connected im kleinen at
x
∈
X
if and only if 2
^{X}
(
K
(
X
),
C_{K}
(
X
)) is connected im kleinen at {
x
} and a locally compact Hausdorff space
X
is connected im kleinen at
x
∈
X
if and only if 2
^{X}
(
C
(
X
),
K
(
X
),
C_{K}
(
X
)) is connected im kleinen at {
x
}. In 2003, Makuchowski
[9
,
10]
investigated with respect to local connectedness at a subcontinuum of continua.
The purpose of this paper is to investigate the relationships between the space
X
and the hyperspace
C
(
X
) concerning admissibility and connectedness im kleinen.
For notational purposes, small letters will denote elements of
X
, capital letters will denote subsets of
X
and elements of 2
^{X}
, and script letters are reserved for subsets of 2
^{X}
. If
B
⊂ 2
^{X}
, ∪
B
= {
A
:
A
∈
B
}. If
A
⊂
X
, the symbol
will denote the interior(closure, boundary) of the set
A
.
X
be a topological space. Let 2
^{X}
= {
E
⊂
X
:
E
is nonempty and closed},
K
(
X
) = {
E
∈ 2
^{X}
:
E
is compact},
C
(
X
) = {
E
∈ 2
^{X}
:
E
is connected}, and
C_{K}
(
X
) =
K
(
X
) ∩
C
(
X
), and endow each with the Vietoris topology. A basis for 2
^{X}
consists of all elements of the form
<
U
_{1}
,
U
_{2}
, ⋯ ,
U_{n}
>= {
A
∈ 2
^{X}
:
A
∩
U_{i}
≠ ø
for each
i
and
where
U
_{1}
,
U
_{2}
, ⋯ ,
U_{n}
are open sets in
X
.
Let
T
(
x
) = {
A
∈
C
(
X
) :
x
∈
A
}. An element
A
∈
T
(
x
) is said to be
admissible
at
x
in
X
if, for each basic open set <
U
_{1}
,
U
_{2}
, ...,
U_{n}
> ∩
C
(
X
) containing
A
, there is a neighborhood
V_{x}
of
x
in
X
such that whenever
y
∈
V_{x}
there is an element
B
∈
T
(
y
) such that
B
∈<
U
_{1}
,
U
_{2}
, ...,
U_{n}
> ∩
C
(
X
)
[11]
.
The space
X
is said to be
locally connected
at
x
in
X
, if for each neighborhood
U
of
x
there is a connected neighborhood
V
of
x
such that
V
⊂
U
[7]
. The space
X
is said to be
connected im kleinen
at
x
, if for each neighborhood
U
of
x
there is a component of
U
which contains
x
in its interior
[7
,
9]
. The space
X
is said to be
locally connected
provided that
X
is locally connected at each of its points. If a space
X
is connected im kleinen at each of its points, then
X
is locally connected. The space
X
is said to be
locally arcwise connected at x
, if for each neighborhood
U
of
x
there is an arcwise connected neighborhood
V
of
x
such that
V
⊂
U
. The space
X
is said to be
locally arcwise connected
, if
X
is locally arcwise connected at each of its points. The space
X
is said to be
arcwise connected im kleinen
at
x
, if for each neighborhood
U
of
x
there is an arcwise connected, component of
U
which contains
x
in its interior. If a space
X
is arcwise connected im kleinen at each of its points, then
X
is locally arcwise connected.
A continuum
X
is said to be
connected im kleinen
at a subcontinuum
A
, if for each open subset
U
of
X
containing
A
, there is a subcontinuum
K
such that
A
⊂
IntK
⊂
K
⊂
U
[10]
. A continuum
X
is said to be
locally connected
at a subcontinuum
A
, if for each open subset
U
of
X
containing
A
, there is an open connected subset
V
such that
A
⊂
V
⊂
U
[1]
. Obviously, if a subcontinuum is degenerate, then the notion of connectedness im kleinen(local connectedness) at a subcontinuum is the same as the notion of connectedness im kleinen(local connectedness) at a point. Note that if
X
is connected im kleinen(locally connected) at each point of
A
, then
X
is connected im kleinen(locally connected) at a subcontinuum
A
, but not conversely,
Result 1.1
(
[12]
). (Boundary Bumpping Theorem)
Let X be a Hausdorff continuum, and let A
∈
C
(
X
).
Then for each open set U in X containing A, the component C_{A} of
containing A intersects Bd
(
U
).
Theorem 2.1.
Let X be a Hausdorff continuum, and let A
∈
C
(
X
).
If for each open set U containing A there is a continuum K and a neighborhood V of a point of A such that V
⊂
IntK
⊂
K
⊂
U, then C
(
X
)
is connected im kleinen at A
.
Proof.
Let
U
=<
U
_{1}
, ⋯ ,
U_{n}
> ∩
C
(
X
) be an open subset of
C
(
X
) containing
A
. Then
is an open subset of
X
containing
A
. And, there is a continuum
K
and a neighborhood
V
of a point
x
of
A
such that
V
⊂
IntK
⊂
K
⊂
U
.
And
Let
L
_{1}
,
L
_{2}
∈<
U
_{1}
, ⋯ ,
U_{n}
,
V
> ∩
C
(
X
). Then
L
_{1}
∩
K
≠ ø and
L
_{2}
∩
K
≠ ø. It follows that
L
_{1}
∪
L
_{2}
∪
K
∈<
U
_{1}
, ⋯ ,
U_{n}
,
V
> ∩
C
(
X
). Hence there is order arcs
L
_{1}
and
L
_{2}
in <
U
_{1}
, ⋯ ,
U_{n}
,
K
> ∩
C
(
X
) from
L
_{1}
to
L
_{1}
∪
L
_{2}
∪
K
and from
L
_{2}
to
L
_{1}
∪
L
_{2}
∪
K
. It follows that there is an arc in
L
_{1}
∪
L
_{2}
from
L
_{1}
to
L
_{2}
, and it is clear that
L
_{1}
∪
L
_{2}
⊂<
U
_{1}
, ⋯ ,
U_{n}
,
V
> ∩
C
(
X
). Therefore
C
(
X
) is
locally arcwise connected
at
A
. ☐
Theorem 2.2.
Let
A
∈
C
(
X
). If
IntA
≠ ø, then for each open set
U
containing
A
there is a continuum
K
and a neighborhood
V
of a point of
A
such that
V
⊂
IntK
⊂
K
⊂
U
.
Proof.
Let
U
be an open set containing
A
. Let
x
∈
IntA
. Then there is an open set
V
such that
x
∈
V
⊂
IntA
, and hence
x
∈
IntA
⊂
A
⊂
U
. In this case
A
is a continuum which satisfies the condition of the continuum
K
in this theorem. ☐
We get the below Corollary from Theorem 2.1 and Theorem 2.2.
Corollary 2.3
([Theorem 3 of
[4]
]). Let
A
∈
C
(
X
). If
IntA
≠ ø,
then C
(
X
)
is locally arcwise connected at A
.
Proof.
Let
A
∈
C
(
X
) and let <
U
_{1}
, ⋯ ,
U_{n}
< ∩
C
(
X
) be a basic open set containing
A
. Let
x
∈
IntA
and let
V
be an open set such that
x
∈
V
⊂
IntA
and such that
Then
A
∈<
U
_{1}
, ⋯ ,
U_{n}
,
V
>⊂<
U
_{1}
, ⋯ ,
U_{n}
>. Let
L
_{1}
,
L
_{2}
∈<
U
_{1}
, ⋯ ,
U_{n}
,
V
> ∩
C
(
X
). Then
L
_{1}
∩
V
≠ ø and
L
_{2}
∩
V
≠ ø, so
L
_{1}
∩
A
≠ ø and
L
_{2}
∩
A
≠ ø. It follows that
L
_{1}
∪
L
_{2}
∪
A
∈<
U
_{1}
, ⋯ ,
U_{n}
,
V
> ∩
C
(
X
). Hence there is order arcs
L
_{1}
and
L
_{2}
in <
U
_{1}
, ⋯ ,
U_{n}
,
V
> ∩
C
(
X
) from
L
_{1}
to
L
_{1}
∪
L
_{2}
∪
A
and from
L
_{2}
to
L
_{1}
∪
L
_{2}
∪
A
. It follows that there is an arc in
L
_{1}
∪
L
_{2}
from
L
_{1}
to
L
_{2}
, and it is clear that
L
_{1}
∪
L
_{2}
⊂<
U
_{1}
, ⋯ ,
U_{n}
,
V
> ∩
C
(
X
). ☐
Theorem 2.4.
Let X be a Hausdorff continuum, and let A
∈
C
(
X
).
If X is connected im kleinen at A, then A is admissible.
Proof
. Let
x
∈
A
∈
C
(
X
) and
X
is
connected im kleinen
at
A
. Let <
U
_{1}
, ⋯ ,
U_{n}
> ∩
C
(
X
) be a basic open set containing
A
, and let
. Then
A
⊂
U
and there is a continuum
K
such that
A
⊂
IntK
⊂
K
⊂
U
. Set
V_{x}
=
IntK
. Then for every
y
∈
V_{x}
,
y
is an element of
K
. And since
A
⊂
K
and
K
⊂
U
,
K
∈<
U
_{1}
, ⋯ ,
U_{n}
> ∩
C
(
X
). Thus
A
is admissible. ☐
Theorem 2.5.
Let X be a Hausdorff continuum, and let A
∈
C
(
X
).
If A is admissible, then for any open subset U of C
(
X
)
containing A, there is an open subset V of X such that
.
Proof.
Let
U
be an open set containing
A
in
C
(
X
), and let
x
∈
A
. Then by the definition of admissibility there is an open set
V_{x}
containing
x
in
X
such that for every
y
∈
V_{x}
there is a continuum
B
in
C
(
X
) such that
y
∈
B
∈
U
. Set
. Then
.
Theorem 2.6.
Let X be a Hausdorff continuum, and let A
∈
C
(
X
).
If for any open subset U of C
(
X
)
containing A, there is a subcontinuum K of X such that A
∈
IntK
⊂
K
⊂
U and there is an open subset V of X such that
then A is admissible.
Proof.
Let
U
=<
U
_{1}
, ⋯ ,
U_{n}
> ∩
C
(
X
) be a basic open subset of
C
(
X
) containing
A
, let
K
a continuum in
C
(
X
) contains
A
in its interior, let
V
an open subset of
X
such that
. Then for any element
y
of
V
,
is a continuum in
U
containing
y
.

0. INTRODUCTION

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1. PRELIMINARIES

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2. CONNECTEDNESS IM KLEINEN AND ADMISSIBILITY

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Citing 'A NOTE ON CONNECTEDNESS IM KLEINEN IN C(X)
'

@article{ SHGHCX_2015_v22n2_139}
,title={A NOTE ON CONNECTEDNESS IM KLEINEN IN C(X)}
,volume={2}
, url={http://dx.doi.org/10.7468/jksmeb.2015.22.2.139}, DOI={10.7468/jksmeb.2015.22.2.139}
, number= {2}
, journal={Journal of the Korean Society of Mathematical Education Series B The Pure and Applied Mathematics}
, publisher={Korean Society of Mathematical Education}
, author={BAIK, BONG SHIN
and
RHEE, CHOON JAI}
, year={2015}
, month={May}