The Necessary Condition For Fixed Points In The Inverse Limits Spaces

A point p in the inverse limit space is said to be a cut point of this space when excluded from it, when the number of the components of that space increases. Therefore, this study aims at finding the necessary condition for fixed points in the inverse limit space to be cut points. Then, for applying the main theorem with some conditions, a sequence of upper semi continuous can be employed as a bonding function to get a union of continua as a generalized inverse limit space if there is a generalized inverse limit for each of them separately.


Introduction
In 2004, Mahavier introduced a generalized inverse limit on intervals [1].Later, in 2006, Ingram and Mahavier introduced this limit on the compact sets [2].Recently, researchers published a number of results about some continuum properties in an inverse limit space.In 2015, Banic and Martinez found the universal dendrite D 3 as the generalized inverse limit space [3].In 2022, Corona studied dendrites as the generalized inverse limit space [4], while Marsh studied atriodic tree like continua as inverse limits on [0,1] [5].Currently, the generalized inverse limit is a powerful tool in the study of continuum theory.
A topological space X is said to be continuum if it is a nonempty, connected, compact and metric space.A subcontinuum is a subset of the continuum.In this regard, 2 X denotes the hyperspace of X when X is a continuum.
A set valued function f: X → 2 Y is said to be an upper semi-continuous function if for each element x in the space X and all open subsets V in the space Y, which contains f(x), there is an open set U in X which contains x such that for each element t in U, then f(t) ⊆ V.If X and Y are compact metric spaces and f: X → 2 Y is a set valued function, then f is an upper semi-continuous function if and only if its graph G(f) = {(x, y): y ∈ f(x)} is a closed subset in X × Y [6, p. 3].Let X and Y be compact Hausdorff metric spaces and f: X → Y be a continuous function.The function f is said to be monotone if for each y ∈ Y the inverse image of y (f −1 (y)) is a continuum.Let (X i ) i∈N be a sequence of continua and f i : X i+1 → 2 X i be an upper semi-continuous function.The generalized inverse limit space of a sequence {X i , f i } is denoted by lim ⟵ {X i , f i } and defined by lim )for all i ∈ N}.All inverse limits in this study are generalized inverse limit spaces.The distance between elements x and y in the inverse limit space is defined by d(x, y) = ∑ , when x = (x 1 , x 2 , . . . ) and y = (y 1 , y 2 , . . . ) are elements in lim ⟵ {X i , f i } and d i is a metric space on X i bounded by 1.More information about inverse limits of continua having set valued upper semi-continuous bonding functions defined on [0,1] can be founded in [7] and [6].

Definition 2.1:
The Gehman Dendrite of order n is denoted by G n , defined by a dendrite such that all of its ramification points are of order n and the set of end points E(G n ) is homeomorphic to the Cantor ternary set.

Definition 2.2:
Harmonic fan continuum is a continuum defined by a union of arcs, joining the point (0,1) to

Definition 2.5:
The Hilbert cube is a continuum which is homeomorphic to the product is less than ϵ.
Definition 2.7: Let X be a continuum and P be a topological property.X is said to be P like if there exists an ϵ map from X to a continuum having the property P.

Definition 2.11:
The Warsaw circle continuum is a union of a continuum X and a continuum Y where X is the topologist's sine curve and Y is an arc joining points (1, sin (1)) and (0, −1).It is a circle like continuum [8].A double Warsaw circle is a union of a double topologist's sine curve continuum and an arc as shown in Figure 2.   Sierpinski universal plane curve is M 1 2 .
Let X o = I 2 .Divide X o into nine congruent squares and remove the middle one to get ]).
Similarly, for the rest of the remaining eight squares to get X 2 .This process is continued in this way to get X 3 , X 4 , ....
The intersection X = ⋂ i=1 ∞ {X i } of all X i , i = 1,2, ... is said to be Sierpinski Universal Curve.It is a locally connected continuum curve and it does not have any cut points.It is embedded in I 2 so that R\X has these components Definition 2.14: A. topological space X is said to be a locally connected continuum if for each element p in X and each neighborhood U of p, there exists a continuum neighborhood of p in U [11].

Definition 2.15:
A dendrite X is said to be a locally connected continuum if it does not have any simple closed curve.Dendrites are hereditary unicoherent that is the intersection of any of its two sub continua is a continuum.
Definition 2.16: Let f: X → X be a continuous function.A point p in X is said to be a fixed point if p = (a, a) is an element in the graph G(f) in X 2 [12].
Definition 2.17: Let f: I → 2 I be an upper semi continuous function and X = lim ⟵ {I, f} be the generalized inverse limit space.A point p in X is said to be a fixed point if p = (a, a, a, a, a, … ) where a ∈ I and a = f −1 (a).

Main Theorems
This section clarifies how the fixed points in the inverse limit space are considered cut points under some The first main theorem in this study is as follows: is a continuum, where f i | [a i−1 ,a i ] : J i = [a i−1 , a i ] → 2 J i , i = 1, . . ., n − 1 is the restriction of f on J i , G(f i ) ∩ y i = (a i , a i ) and G(f i ) ∩ x i = (a i , a i ) where y i (x) = a i and x i (y) = a i are horizontal and vertical line segments, respectively.
If y i (x) ∩ G(f i ) is a non-degenerate, then y i (x) ∩ G(f k ) is degenerate for k ≠ i.If the inverse limit is a continuum and points p i = (a i , a i , a i , . . .Proof.Since for each t ∈ int(J i ), f −1 (t) ∈ int(J i ), so for each t ∈ f i (J i ) ∩ int(J i ), f −1 (t) ∈ f i (J i ) ∩ int(J i ).So, G(f i ) and G(f i −1 ) are subsets of I i 2 .Let (x, y) ∈ int(G(f i )).It is clear from the definition of f i that a i−1 < x, y < a i . Let

Example 2 . 3 :Definition 2 . 4 :
, n ∈ N together with the arc A = {(0, y),0 ≤ y ≤ 1}.It is not a locally connected continuum because all points in the limit bar are non-locally connected points.It is not a dendrite because it is not locally connected.The continuum F ω is a dendrite defined by the union of sequence of straight lines {l n } n+1 ∞ such that ⋃ n=1 ∞ {l n } has only one ramification point of order ω and lim n→∞ H d (l n , 0) = 0,[7] and[8].Let m ∈ {3,4, . . ., ω}, the universal dendrite of order m and be denoted by D m such that all of its ramification points are of order m and for each arc subset A ⊂ D m , the set of ramification points in the dendrite D m located in A is dense in A.

Figure 2 Definition 2 . 12 : 2 , 0 ) 1 ]
Figure 2 Warsaw circle with two limit bars Definition 2.12: The Knaster, BJK or buckethandle continuum is denoted by K as shown in Figure 3 and defined by the following: the non-negative ordinal set of all semi circles with ( 1 2 , 0) center and intersect Cantor set C; the non-positive ordinal set of all semi circles such that ∀n ∈ N, with center ( 5 2.3 n , 0) and intersect all Cantor set points

Figure 4 :
Figure 4: A Double Knaster continuum Definition 2.13: The Menger continua represent a universal continuum M n m , 1 ≤ m ≤ n, and defined as follows: restrictions.It starts with some basic definitions.A point p in a dendrite D is said to be an endpoint of the dendrite D if for any two arcs containing p there is another point in the intersection of them.The point p in the dendrite D is an ordinary point of D if D ∩ {p} c has only two components, and the point p is said to be a ramification point of the dendrite D if D ∩ {p} c has n components for n ≥ 3. The order of a point p in a dendrite D is n, where n is an element in the set N ∪ {ω}, if D ∩ {p} c has n components.These notations are used: E(D) is used for the set of end points of the dendrite D and R(D) is used for the set of ramification point of D. The dendrite G n or Gehman dendrite or order n is the dendrite where all of its ramification points are of order n and its E(D) is homeomorphic to the Cantor set [13, Theorem 4.1].