Making Holes in the Second Symmetric Product of a Cyclicly Connected Graph

A continuum is a connected compact metric space. The second symmetric product of a continuum X, F2(X), is the hyperspace of all nonempty subsets of X having at most two elements. An element A of F2(X) is said to make a hole with respect to multicoherence degree in F2(X) if the multicoherence degree of F2(X)− {A} is greater than the multicoherence degree of F2(X). In this paper, we characterize those elements A ∈ F2(X) such that A makes a hole with respect to multicoherence degree in F2(X) when X is a cyclicly connected graph.


Introduction
A continuum is a connected compact metric space.Let X be a continuum.For each positive interger n, let F n (X) = {A ⊂ X : A has at most n elements and A ∅}.The hyperspace F n (X) is called the n th symmetric product of X.It is known that each hyperspace F n (X) is a continuum (see Borsuk & Ulam, 1931, pp. 876, 877) and (Michael, 1951, Theorem 4.10, p. 165).
If Z is any topological space, let b 0 (Z) denote the number of components of Z minus one if this number is finite and b 0 (Z) = ∞ otherwise.Given a connected topological space Y, the multicoherence degree of Y, is defined by r(Y) = sup{b 0 (K ∩ L) : K and L are closed connected subsets of Y and Y = K ∪ L}.The space Y is said to be unicoherent if r(Y) = 0. Let y ∈ Y such that Y − {y} is connected, we say that y makes a hole with respect to multicoherence degree in Y if r(Y − {y}) > r(Y).This is a generalization of the notion of to make a hole in a unicoherent topological space defined in (Anaya, 2007(Anaya, , p. 2000)).
In this paper, we are interesting in the following problem.
Problem.Let H(X) be a hyperspace of a continuum X.For which elements A ∈ H(X), A makes a hole with respect to multicoherence degree in H(X).
In the current paper, we are presenting the solution to this problem when X is a cyclicly connected graph and H(X) = F 2 (X).

Preliminaries
Given a positive interger m, define λ(m) = {1, 2, . . ., m}.A map is a continuous function.The identity map for a topological space Z is denoted by id Z .An arc is any space homeomorphic to [0, 1].A simple closed curve is a space which is homeomorphic to the unit circle S 1 in the Euclidean plane R 2 .A theta curve is a space which is homeomorphic to S 1 ∪ ([−1, 1] × {0}) in R 2 .The symbol [0, 1] 2 denotes the space [0, 1] × [0, 1].The set {(u, v) ∈ [0, 1] 2 : u ≤ v} is denoted by Δ.A graph is a continuum which can be written as the union of finitely many arcs any two of which are either disjoint or intersect only in one or both their end points.A point y in a connected topological space Y is called cut point (non-cut point) if Y − {y} is not connected (connected).A space W is said to be cyclicly connected provided that every two points of W belong to some simple closed curve in W (see (Whyburn, 1942, p. 77)).A graph X is a cyclicly connected graph if X is a cyclicly connected space.
Given a topological space Y.A subspace Z of Y is said to be: (c) a strong deformation retract of Y if there exist f and g as in (b) with the additional property that g(z, t) = z for every (z, t) Let y ∈ Y. Let β be a cardinal number.We say that y is of order less than or equal to β in Y, written ord(y, Y) ≤ β, provided that for each open subset U of Y containing y, there exists an open subset V of Y such that y ∈ V ⊂ U and the cardinality of the boundary of V is less than or equal to β.We say that y is of order β in Y, written ord(y, Y) = β, provided that ord(y, Y) ≤ β and ord(y, Y) α for any cardinal number

Auxiliary Results
Lemma 2.1 If X is a cyclicly connected graph different from a simple closed curve, then the following conditions hold: (1) for each simple closed curve S in X, S ∩ R(X) has at least two points; (2) X = I(X); (3) the set I(X) is finite; (4) for each p ∈ X, M(p, X) is a nondegenerate subcontinuum of X.
Proof.In order to prove (1), let S be a simple closed curve in X.Since S X, there exists a simple closed curve S 1 S in X such that S ∩ S 1 ∅.So, using (Nadler, Jr., 1992, Proposition 9.5, p. 142), R(S ∪ S 1 ) ∩ S ∩ S 1 ∅.Thus, by (Kuratowski, 1968, Theorem 3, p. 278), R(X) ∩ S ∩ S 1 ∅.Now, assume that R(X) ∩ S ∩ S 1 consists of precisely one point.Then, there exists a simple closed curve S 2 S in X such that S 2 ∩ (S − S 1 ) ∅. Applying the previous argument to S ∪ S 2 , we have R(X) ∩ (S − S 1 ) ∩ S 2 ∅.Hence, S ∩ R(X) has at least two points.
Finally, to check (4), let p ∈ X.By (2), there exists I ∈ I(X) such that p ∈ I. So, since I ⊂ M(p, X), M(p, X) is nondegenerate set.On the other hand, clearly, M(p, X) is connected.By (3), M(p, X) is closed in X.
Lemma 2.2 Let X be a cyclicly connected graph and let p ∈ X.If N(p, X) ∅, then N(p, X) is a subcontinuum of X.
Proof.First, by (3) of Lemma 2.1, N(p, X) is closed in X.We shall prove the connectedness of N(p, X).By (Whyburn, 1942, (9.3) Since N(p, X) ∩ M(p, X) = F and by the definition of f , f is well defined.Clearly, f is surjective.The continuity of f follows from the continuity of f and the fact that N(p, X) and M(p, X) − {p} are closed subsets of X − {p}.This finishes the proof of that N(p, X) is connected.
Lemma 2.3 Let X be a cyclicly connected graph different from a simple closed curve and let p, q be different points in X.If X − {p, q} is not connected, there exist a simple closed curve S in X containing p and q and a retract f : Given I ∈ I(X), let f I : I → S be a one-to-one map such that Define f : X → S as follows: for each x ∈ X, take Hence, f is well defined.The continuity of f follows from the fact that each f I is continuous and, by ( 2) and (3) of Lemma 2.1.It is easy to see that f | S = id S .Thus, f is a retraction.
From the fact that p q, we have that f −1 (p) = {p} and f −1 (q) = {q}.
Lemma 2.4 Let X be a cyclicly connected graph different from a simple closed curve and let p, q be different points in X.If X − {p, q} is connected, there exist a theta curve Y in X containing p and q and a retract f : Proof.By the definition of cyclic connectedness, there exists a simple closed curve S in X such that p, q ∈ Y. Since X − {p, q} is connected, there exists an arc J in X such that S − {p, q} Define f : X → Y as follows: for each x ∈ X, take I ∈ I(X) such that x ∈ I and let f (x) = f I (x).From the fact that f | R(X) = f 0 , it follows that f is well defined.Since X = I(X) and I(X) is finite (see ( 2) and (3) of Lemma 2.1), f is continuous.From the fact that f | Y = id Y , it follows that f is a retraction.
We will prove that f Proposition 2.5 Let X be a continuum and let K and L be connected subsets (subcontinua) of X.Then K, L is a connected subset (subcontinuum) of F 2 (X) and, it does not have cut points when K and L are nondegenerate sets.
In order to prove the second part of this proposition, let {p, q} ∈ K, L .Using K and L are nondegenerate sets and the arguments in (Kuratowski, 1968, Theorem 11, p. 137), it can be shown that Proof.
It is easy to verify that f and g have the required properties.
Finally, let h: [0, 1] → I be a homeomorphism such that h([0, It can be proved that h is a homeomorphism such that h(Γ 0 ) = H, I ∪ J, I .Therefore, H, I ∪ J, I is a strong deformation retract of F 2 (I) − {{p}}.
Lemma 2.7 If X is a graph containing a simple closed curve, then X is not unicoherent.
Proof.We shall prove that there exist subcontinua K and L of X such that b 0 (K ∩ L) > 0 and X = K ∪ L. Let S be a simple closed curve in X.By (Nadler, Jr., 1992, Theorem 9.10, p. 144), there exists x ∈ S such that ord(x, X) = 2. Now, using (Nadler, Jr., 1992, Theorem 9.7, p. 143), it can be proved that there exists an arc J in S which is a neighborhood of x in X.Then, J − E(J) is an open connected subset of X.Now, by (Nadler, Jr., 1992, 9.44, (a), p. 160), S −(J −E(J)) is connected.Hence, X −(J −E(J)) is a subcontinuum of X.So, K = J and L = X −(J −E(J)) satify the requiered properties.

Making Holes in the Second Symmetric Product of a Cyclicly Connected Graph
Theorem 3.1 Let X be a graph and let p ∈ O(X).Then {p} does not make a hole with respect to multicoherence degree in F 2 (X).
Since p ∈ O(X), using (Nadler, Jr., 1992, Lemma 9.7, p. 143), it can be shown that there exists an arc I in X such that I is a neighborhood of p in X.So, clearly, p ∈ I − E(I).Let H and J be nondegenerate subcontinua of I such that H ∪ J ⊂ I −{p} and each one of them contains a different end point of I. Put Z = (X − I)∪ H ∪ J and Z = X, Z .Clearly, F 2 (X) = Z ∪ F 2 (I).Now, by Lemma 2.6, there exist a retraction f : F 2 (I) − {{p}} → H, I ∪ J, I and a map g: (F 2 (I) − {{p}}) × [0, 1] → F 2 (I) − {{p}} such that g(A, 0) = A and g(A, 1) = f (A) for each A ∈ F 2 (I) − {{p}} and g(B, t) = B for each (B, t) ∈ ( H, I ∪ J, I ) To check that f and ḡ are well defined, notice that Z ∩ F 2 (I) . Now, the continuity of f and ḡ follows from the continuity of the maps f and g and the fact that Z and F 2 (I) − {{p}} are closed in F 2 (X) − {{p}}.It is easy to verify that f and ḡ have the required properties.Thus, Z is a deformation retract of F 2 (X) − {{p}}.
Finally, to check that r(Z) = r(F 2 (X)), we shall show that Z is homeomorphic to F 2 (X).It can be shown that there exists a homeomorphism h: It is easy to see that h is a homeomorphism.Hence, r(F 2 (X)) = r(Z).
This finishes the proof that {p} does not make a hole with respect to multicoherence degree in F 2 (X).
Theorem 3.2 Let X be a cyclicly connected graph and p ∈ R(X).Then {p} makes a hole with respect to multicoherence degree in F 2 (X).
Proof.Since r(F 2 (X)) = 1 (see Theorem 2.8), we shall show that r(F 2 (X) − {{p}}) ≥ 2. So, it suffices to prove that there exist two closed connected subsets K and L of F 2 (X) − {{p}} such that F 2 (X) and, ϕ k and ϕ j are one-to-one, ψ (k, j) is well defined.Using the fact that ϕ k and ϕ j are surjective, it is easy to prove that ψ (k, j) is surjective.Clearly, for each k, j ∈ λ(m) with k j, Consider the following cases.
We are ready to prove that K ∩ L = {C k : k ∈ λ(m)}.From the fact that Σ = Λ ∩ Γ and ii), we have that . This case can be proved using similar arguments in the proof of Case A by considering Y = {ϕ 1 (1)}.
Theorem 3.3 Let X be a simple closed curve and let p, q ∈ X such that p q. Then {p, q} makes a hole with respect to multicoherence degree in F 2 (X).
We are ready to prove that {p, q} makes a hole with respect to multicoherence degree in F 2 (X).Since X is a simple closed curve, there exists a homeomorphism h: S 1 → X such that h(A) = {p, q}.Consider the induced mapping h 2 : F 2 (S 1 ) → F 2 (X) defined by h 2 (B) = h(B) for each B ∈ F 2 (S 1 ).By (Higuera & Illanes, 2011, Theorem 3.1, p. 369), h 2 is a homeomorphism.Then, since A makes a hole with respect to multicoherence degree in F 2 (S 1 ) and h 2 (A) = {p, q}, {p, q} makes a hole with respect to multicoherence degree in F 2 (X).
Theorem 3.4 Let X be a theta curve and let p, q ∈ X such that ord(p, X) = ord(q, X) = 2 and X − {p, q} is connected.Then {p, q} makes a hole with respect to multicoherence degree in F 2 (X).

Lemma 2. 6
Let I be an arc and let p ∈ I − E(I).If H and J are subcontinua of I such that H ∪ J ⊂ I − {p} and each one of them contains a different end point of I, then H, I ∪ J, I is a strong deformation retract of F 2 (I) − {{p}}.