Volume 02 Issue 12-2022
227
International Journal of Advance Scientific Research
(ISSN
–
2750-1396)
VOLUME
02
I
SSUE
12
Pages:
227-235
SJIF
I
MPACT
FACTOR
(2021:
5.478
)
(2022:
5.636
)
METADATA
IF
–
7.356
A
BSTRACT
In this paper, we will show how to teach general topology by using traditional methods with examples and
show some applications of the rules of topology, especially in the concept of continuity between topological
spaces. The application will be given as a definition of society and education through the continuity of
topological spaces.
K
EYWORDS
Topology, the concept of continuity, statements, expressions, the juridical regime of heraldry, the
incorporeal transformations.
I
NTRODUCTION
Definition1. Let
1
( , )
X
and
2
( ,
)
Y
are two
topological spaces. A mapping
f
from to is
called continuous, if
1
1
( )
f
U
−
for every
2
U
, i.e. if the preimage of every open subset of the
Journal
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Copyright:
Original
content from this work
may be used under the
terms of the creative
commons
attributes
4.0 licence.
Research Article
METHOD OF TEACHING CONTINUOUS MAPPING OF
TOPOLOGICAL SPACE AND ITS APPLICATION BY THE
CONCEPT OF SOCIETY AND EDUCATION
Submission Date:
December 19, 2022,
Accepted Date:
December 24, 2022,
Published Date:
December 29, 2022
Crossref doi:
https://doi.org/10.37547/ijasr-02-12-32
Shakhnoza Sodikkhujaeva
Master Of Science, Assistant, Higher Mathematics Department, Fergana Polytechnic Institute, Fergana,
Uzbekistan
Volume 02 Issue 12-2022
228
International Journal of Advance Scientific Research
(ISSN
–
2750-1396)
VOLUME
02
I
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12
Pages:
227-235
SJIF
I
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FACTOR
(2021:
5.478
)
(2022:
5.636
)
METADATA
IF
–
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space is an open subset of the space
X
. We
denote continuous mapping
f
of the space
X
to
Y
in the form
:
f X
Y
→
.
Example 1.
1.
Let
: ( ,
)
( , )
D
f
X
Y
→
−
continuous
mapping,
as
V
follows
1
1
( )
( )
D
f
V
X
f
V
−
−
,
where
D
−
discrete topology in
X
and
−
any
topology in
Y
.
2.
Let
: ( , )
( ,
)
A
f
X
Y
→
−
continuous
mapping, as
A
V
follows
V
Y
=
or
V
=
Hence
1
( )
f
Y
X
−
=
or
1
( )
f
−
=
, where
A
−
indiscrete
topology in
Y
and
is any topology in
X
.
3.
Let
{ , , }
X
a b c
=
,
1
{ ,
,{ },{ },{ , }}
X a
b
a b
=
and
{ , }
Y
x y
=
,
2
{ , ,{ }}
Y x
=
.
Let
1
2
)
)
: ( ,
( ,
f
X
Y
→
and
( )
( )
, ( )
f a
f b
x f c
y
=
=
=
.
Then
f
−
continuous,
as
2
Y
consequently
1
( )
f
Y
X
−
=
,
2
,
1
( )
f
−
=
,
2
{ }
x
,
1
({ }) { , }
f
x
a b
−
=
.
Example 2.
Prove, that a mapping
:
f R
R
→
,
definition of
−
continuity is followed from the
definition of an open mapping.
Let
:
f R
R
→
is continuous in the point
x
R
, if
for every
0
, there exists
0
such that,
( )
( )
f y
f x
−
for every
y
, where
y
x
−
.
Then
f
is continuous, if it is continuous in every
point
p
R
.
Proof.
Suppose that
:
f R
R
→
is continuous by
definition
−
. We will show that from the
property (4) in preposition [1]
f
is continuous
for open subsets. Let's consider any point
x
R
and any neighbourhood
V
that contains
( )
f x
.
There exists basis element
( , )
c d
contains
( )
f x
,
such
that
( , )
c d
V
.
Let
min( ( )
, ( )
)
f x
c f x
c
=
−
+
, notice that
0
as
( )
c f x d
.
It
is
trivial
to
( ( )
, ( )
)
( , )
f x
f x
c d
V
−
+
and contains
x
.
Since
f
is continuous in the point
x
, there exists
0
such that
y
x
−
sequently
( )
( )
f y
f x
−
for every
y
. Let
(
,
)
U
x
x
= −
+
, it is clear, that
it is the neighbourhood of the point
x
. Let's
consider any point
( )
z
f U
such that
( )
z
f y
=
for some
y U
. Let then we have
x
y x
− +
such that o
y
x
− −
, therefore it is followed by
y
x
−
We know that
( )
( )
( )
z
f x
f y
f x
−
=
−
such that
f
is continuous. Hence
( )
z
f x
− −
such that
( )
( )
f x
z f x
−
+
, and consequently
z
V
as
(
( ( )
,
)
)
f x
f x
V
−
+
.
( )
z
f U
is
arbitrary, and this shows that
( )
f U
V
, and
from the property (4) is fulfilled for
f
as
x
was
arbitrary.
Definition 2.
Continuous mapping
:
f X
Y
→
is
called homeomorphism, if
f
bijective maps the
space
X
onto
Y
and inverse image
1
f
−
from
Y
to
X
is continuous.
Two topological spaces
X
and
Y
are called
homeomorphic or topological if there exists
Volume 02 Issue 12-2022
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VOLUME
02
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)
METADATA
IF
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homeomorphism of the space
X
onto the space
Y
and denoted by
X
Y
.
Example 3.
Let denote by
X
and
*
X
single tone
sets with two topologies
and
*
, respectively.
Let
*
:
i
→
is an identity mapping.
1.
We will show that
i
is continuous, if only if,
when
*
is weaker than
.
2.
We will show that
i
is a homeomorphism, if
only if, when, when
*
=
.
Solution: 1) Notice that the inverse of an identity
mapping is a domain, thus it is an inverse for any
subset
*
A
X
X
=
, so we have
1
( )
( )
i A
i
A
A
−
=
=
.
(
⇒
) Suppose that
i
is continuous and consider
any open subset
U
. Then we have
1
( )
i
U
U
−
=
an open subset
*
as
i
is continuous. Since
U
is
arbitrary, it shows that
*
consequently
*
is
weaker. (
⇐
) Let's consider that
*
is weaker, as
*
. Consider any open set
U
such that
*
U
, i.e.
U
is also open in
*
. Since
1
( )
i
U
U
−
=
this shows that
i
is continuous by the definition
of continuity. 2) It is obvious, that
i
is objective as
domain and image are the same i.e.
1
i
i
−
=
.
i
is a
homeomorphism, if only if, when
i
and
1
i
−
are
continuous. Sequently
*
is weaker than
, and
is weaker than
( from (1)), hence
*
consequently
*
=
In topology, elements
function in multiple surfaces and are affected by
each surface. In this way, topology changes our
understanding of “familiar social science objects
of research by mapping out how such objects
change and how they relate, in this process, to
other changing objects in multiple, relational
spaces”
. Multiplicity means that elements belong
to more than one surface [1-7]. An element that is
part of two topologies is the site of three effects,
the effects of each surface and the effects of the
surfaces formed by the combination of both
surfaces. This does not mean that all the
topologies active at some point are equally
affective at all times. While elements of scientific
management and education for efficiency may
still resonate, there have been certain spaces and
time where these elements were more affective.
Another important attribute is that topologies
continuously, though not always rapidly, change
their internal dynamics, or deform. In our
increasingly topological society, movement
–
as
the ordering of continuity
–
composes the forms
of social and cultural life themselves [8-19]. This
is not a matter of one rationality displacing the
other, but of their overlapping and mutual
implication such that the continuity of movement
–
or the continuum
–
becomes fundamental to
contemporary culture. This continuous multiple
deformation occurs across axes that are
immanent to a topology and so deformation
occurs along certain ‘lines’ or according to certain
principles. The famous deformation of a coffee
cup into a doughnut is possible because it follows
the line of ‘having only one hole’. But topology
also theorizes fuzzier, yet mathematically
rigorous, ‘shape consistency’ under deformation.
This can usefully be compared to other things that
change yet are held to remain the same, such as a
family or community or group
–
vitalities, that is,
intangible-but-real-entities that remain despite
turnover in membership. In topological thinking,
unless one of the co-constitutive surfaces is
ruptured, and its topological relation is lost, all
Volume 02 Issue 12-2022
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VOLUME
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topologies continue to produce effects [20-27].
So, unlike the virtual, which is beyond experience
and the experienced actual, topological figures
cut across the distinction of the virtual and actual.
The movement, the process at stake is not the
generation of an actual by a virtual, but the
deformation of, as it were, two actuals into one
another via their topological properties.
Topological surfaces enfold and re-enfold each
other in a complicated dance of continuously
deforming multiplicity. The main questions
concern the topological principles at work on an
element and the ways they are inflected and
deflected by other topological principles at work
on and through that element [28-36]. The surface
formed by, the interaction between two or more
topologies is an effect of the multiple binding
principles at work in the various surfaces of
which an element is a part. The immanence of the
principles at work in every topological surface,
though, means that the effects of the
deformations of an element belonging to multiple
surfaces are not found. ‘Interior’ and ‘exterior’
make a different sense in topological thinking, as
insides and outsides are continuous … borders of
inclusion and exclusion do not coincide with the
edges of a demarcated territory, and it is the
mutable quality of relations that determines
distance and proximity, rather than a singular and
absolute measure. In a topological society, the
nation is not necessarily bigger or stronger than,
say, an electricity meter, and the domestic is not
necessarily situated at a lower level than a map of
the world. The education topological function at
work in the assemblage is made up of multiple
constant connections of topologies; architectural
topologies, individualizing surfaces, amassing
topologies, knowledge topologies, semiotic
surfaces, corporeal topologies, subjectivizing
topologies
–
recalling that every element or point
exists in multiple topologies at multiple times.
The ‘teacher’, for example, is actualized through
multiple
topological
affects
(including
comportments, materials, curricula), as it is
constructed by topologies that connect and by the
ways those connections work; “form relate to
populations, populations imply codes, and codes
fundamentally include phenomena of relative
decoding that are all the more usable,
composable, and addable by virtue of being
relative, always beside [37-42]. Co-constitutive
topologies construct this subject ‘teacher’. Space
is multi-dimensional and constituted through
mutually implicated; so, it is not hard to see the
teaching as produced through these topological
effects in the same space at the same time. It is
this co-constitution through multiple topologies
that is crucial because they are coterminous,
overlapping, connecting and continuously
deforming. They are machinic, in being
“simultaneously located at the intersection of the
contents and expression on each stratum, and at
the intersection of all of the strata with the plane
of consistency [43-47]. They rotate in all
directions, like beacons”. Rotation, stretching,
deforming, education is a dynamic topological
affect at the molecular level. This requires
addressing bodies, enunciations and their
relations; for example, mapping is the
“
intermingling of bodies defining feudalism: the
div of the earth and the social div; the div of
the overlord, vassal, and serf; the div of the
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knight and the horse and their new relation to the
stirrup; the weapons and tools assuring a
symbiosis of bodies” as well as those “statements,
expressions, the juridical regime of heraldry, all of
the incorporeal transformations, in particular,
oaths and their variables: the collective
assemblage of enunciation”. Within each
topological assemblage, what is important is the
ways that lines of deterritorialization form points
of intersection between enunciative acts on the
one hand, and the machinic assemblage of bodies,
their attributes, actions and capacities on the
other.
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235
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(ISSN
–
2750-1396)
VOLUME
02
I
SSUE
12
Pages:
227-235
SJIF
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5.636
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METADATA
IF
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