A–SETS AND ABCOHESIVE SPACES

David John

Missouri Western State College

\underline{Definitions}. A space M is abcohesive at a point p with respect to a point q if there exists an open connected set U such that p is a point in U and U is a subset of M − {q}. The space M is abcohesive at a point p if it is abcohesive at p with respect to q for each q in M − {p}. The space M is abcohesive if it is abcohesive at p for each p in M.

\underline{Remarks}. If p is a non–cut point of M, and M is T_{1} then M is abcohesive at each point q in M − {p} with respect to p. Hence, if each point of M is a non–cut point of M, then M is abcohesive. Also, if M is a locally connected T_{1} space, then M is abcohesive. Sierpinski space is locally connected but not abcohesive. However, Sierpinski space is not T_{1}. For the remainder of this paper, we will assume the space M is Hausdorff. If M is a continuum, then there exist two points p and q in M such that M is abcohesive at each x in M − {p} with respect to p and at each x in M − {q} with respect to q.

\underline{Theorem 1}. The space M is abcohesive at each point q in M − \{p\} with respect to p if and only if each component of M − {p} is open.

\underline{Proof}. Suppose M is abcohesive at each point q in M − \{p\} with respect to p. Let C be a component of M − {p}, and let x be a point in C. Since M is abcohesive at x with respect to p, there exists an open connected set K such that x ∈ K and K ⊂ M − {p}. But K ⊂ C, and hence C is open.

If the components of M − {p} are open, then for each q in M − {p}, there exists a component C such that q ∈ C and C ⊂ M − {p}. Therefore, M is abcohesive at q with respect to p.

\underline{Theorem 2}. If M is an abcohesive connected space and C is a component of M − \{p\}, then p is a limit point of C.

\underline{Proof}. Let C be a component of M − \{p\}. If p is not a limit point of C, then C is both open and closed in M. This involves a contradiction. Hence p is a limit point of C.

\underline{Theorem 3}. If M is an abcohesive space, then the components of M are open.

\underline{Proof}. If M is connected, then M is the only component of the space. If M is not connected, then let C be a component of M and let p be a point in M − C. By Theorem 1, the components of M − {p} are open. Since C is a subset of M − {p}, C is a component of M − {p}, and hence C is open.

\underline{Definitions}. A space M is aposyndetic at a point p with respect to a point q if there exists a closed connected set H such that p is in the interior of H and H is a subset of M − {q}. The space M is aposyndetic at a point p if it is aposyndetic at p with respect to q for each q in M − {p}. The space M is aposyndetic if it is aposyndetic at p for each p in M. A space M is semi–locally connected at a point p of M if each open set containing p contains an open set V containing p such that M − V has at most a finite number of components. The space M is semi–locally connected if it is semi–locally connected at each point p in M.

\underline{Theorem 4}. If M is aposyndetic at each point q in M − \{p\} with respect to p, then M is abcohesive at each point q in M − {p} with respect to p.

\underline{Proof}. Let q be a point in M −\{p\} and let C be the component of M −\{p\} containing q. There exists a closed connected set H such that q is in the interior of H and H ⊂ M − {p}. Now H ⊂ C, and hence C is open. Thus M is abcohesive at each point q in M − {p} with respect to p.

\underline{Theorem 5}. If M is an aposyndetic space, then M is an abcohesive space. Jones [1] established Theorem 6.

\underline{Theorem 6}. If the space M is semi–locally connected at p, then M is aposyndetic at each point q of M − {p} with respect to p.

\underline{Theorem 7}. If M is a semi–locally connected space, then M is an aposyndetic space. Theorem 8 follows from Theorems 4 and 6. \underline{Theorem 8}. If the space M is semi–locally connected at p, then M is abcohesive at

each point q in M − {p} with respect to p.

\underline{Theorem 9}. If M is a semi–locally connected space, then M is an abcohesive space. \underline{Theorem 10}. If M is an abcohesive connected space and M has two cut points, then

there exist disjoint closed sets H and K such that M − H and M − K are connected and H and K have non–empty interiors.

\underline{Proof}. Let p and q be cut points of M. Let E be the component of M − \{p\} containing q and let F be the component of M − {q} containing p. Let 𝔄 be the collection of all components of M − {p} different from E and let 𝔅 be the collection of all components of M − {q} different from F. Since M is abcohesive, A and B are open for each A in 𝔄 and B in 𝔅. For each A in 𝔄, A ∪ {p} is a connected subset of M − {q}. Thus A ∪ {p} ⊂ F. Also for each B in 𝔅, B ∪ {q} is a connected subset of M − {p}. Thus B ∪ {q} ⊂ E. Hence A ∪ {p} and B ∪ {q} are disjoint for each A in 𝔄 and B in 𝔅. Let H = ∪𝔄 ∪ {p} and let K = ∪𝔅 ∪ {q}. Now, H and K are disjoint, M − H = E, and M − K = F, and both H and K have non–empty interiors.

It is well–known that every non–degenerate continuum has at least two non–cut points. Theorem 11 is a generalization of this well–known theorem.

\underline{Theorem 11}. If M is a non–degenerate abcohesive connected space, then there exist two disjoint closed connected sets H and K such that H and K have degenerate boundaries and M − H and M − K are connected.

\underline{Proof}. If M contains two non–cut points p and q, then let \{p\} = H and \{q\} = K. If M does not contain two non–cut points, then M has two cut points and Theorem 11 follows from Theorem 10.

\underline{Definition}. Let M be a connected space. An A–set of M is a closed subset of M such that M − A is the union of a collection of open sets each bounded by a single point of A.

\underline{Theorem 12}. If M is an abcohesive connected space and A is a closed set in M, then A is an A–set if and only if for each component C of M − A, C is open and there is a point p of A such that ∂C = {p}.

\underline{Proof}. Let A be an A–set of M. let 𝔘 be a collection of open sets such that M −A = ∪𝔘 and ∂U is a degenerate subset of A for each U in 𝔘. Suppose C is a component of M − A. There exists an element U in 𝔘 such that C ⊂ U. Let ∂U = {p}, and let K be the component of M − {p} containing C. Since M is abcohesive, K is open. K is connected, and U is separated from (M −{p})−U, and so K ⊂ U. Now K is a connected subset of M −A, which implies the component C of M − A must contain K. Hence K = C, and each component of M − A is open and has a degenerate boundary in A.

The converse is easy.

The proof of Theorem 13 follows from the proof of Theorem 12.

\underline{Theorem 13}. If M is an abcohesive connected space, A is an A–set of M, and C is a component of M − A, then for some point p of A, C is a component of M − {p}.

\underline{Theorem 14}. If M is an abcohesive connected space, p ∈ M, and C is a component of M − {p}, then C ∪ {p} is an A–set of M.

\underline{Proof}. The components of M − \{p\} are open and each has boundary \{p\}. Then M − (C ∪ {p}) is the union of all components of M − {p} different from C, and hence C ∪ {p} is an A–set of M.

\underline{Theorem 15}. If M is an abcohesive continuum, A is an A–set of M, and Z is a subcontinuum of M, then Z ∩ A is a continuum.

\underline{Proof}. Assume there exist non–empty separated sets H_{1} and H_{2} such that

Z ∩ A = H_{1} ∪ H_{2} , Z ∩ A ∩ H_{1} ≠ ∅ , and Z ∩ A ∩ H_{2} ≠ ∅.

Now each component of Z −A has a boundary point in Z ∩A. Let ℭ_{1} be the collection of all components of Z − A with at least one boundary point in H_{1}, and let ℭ_{2} be the collection of all components of Z − A with at least one boundary point in H_{2}. Let 𝔎_{1} be the collection of all components of M −A with at least one boundary point in H_{1}, and let 𝔎_{2} be the collection of all components of M − A with at least one boundary point in H_{2}. By Theorem 13, for each component C of M − A, there exists a point x in A such that C is a component of M − {x}. Since M is abcohesive, C is open. Now ∪𝔎_{1} and ∪𝔎_{2} are separated sets. Since each member of ℭ_{1} is in some member of 𝔎_{1} and each member of ℭ_{2} is in some member of 𝔎_{2}, ∪ℭ_{1} is separated from ∪ℭ_{2}. Let Z_{1} = ∪ℭ_{1} ∪ H_{1}, and let Z_{2} = ∪ℭ_{2} ∪ H_{2}. Then Z = Z_{1} ∪ Z_{2}.

Suppose ∪ℭ_{2} is not separated from H_{1}. Since H_{1} is closed, ∪ℭ_{2} ∩ \overline{H_{1}} = ∅. Now there must exist a net N in ℭ_{2} and a point p in H_{1} such that p ∈ lim sup N. Let \overline{N} be the net of closures of the elements of N. \overline{N} is a net of continua and each element of \overline{N} has a point in H_{2}. Since p ∈ lim sup \overline{N}, some subnet of \overline{N} converges to a continuum K containing p. \overline{N} is a net in the compact space Z, and so K ⊂ Z. Since K ∩ H_{1} ≠ ∅ and K ∩ H_{2} ≠ ∅, it follows that K ⊂ A. Now K ⊂ Z ∩ A = H_{1} ∪ H_{2}, and this is a contradiction. Hence ∪ℭ_{2} is separated from H_{1}, and similarly ∪ℭ_{1} is separated from H_{2}. Therefore Z_{1} is separated from Z_{2}, which contradicts the fact that Z is connected. Hence Z ∩ A is connected.

The proof of the following theorem is similar to Whyburn’s proof of this theorem in [2], and is omitted here.

\underline{Theorem 16}. If M is a semi–locally connected continuum and A is an A–set of M, then A is a semi–locally connected continuum.

The next theorem follows from Theorem 15.

\underline{Theorem 17}. If M is an abcohesive continuum, then each A–set of M is a continuum. \underline{Theorem 18}. If M is an abcohesive continuum, A is an A–set of M, a and b are points

in A, and L is an irreducible continuum from a to b, then L ⊂ A.

\underline{Proof}. By Theorem 17, A is a continuum. By Theorem 15, L ∩ A is a continuum, and hence L ∩ A is a subcontinuum of A containing a and b. Therefore, L ∩ A = L and L ⊂ A.

\underline{Theorem 19}. If M is an abcohesive connected space and A is a closed set in M, then A is an A–set of M if and only if each component of M − A has a degenerate boundary.

\underline{Proof}. If A is an A–set, then by Theorem 12, each component of M − A has exactly one boundary point. Let C be a component of M − A, let ∂C = {p}, and let K be the component of M − {p} containing C. Since M is abcohesive, K is open. Now

M − {p} = C ∪ [(M − {p}) − C],

C is separated from (M −{p})−C, and K is a connected subset of M −{p}, and so K ⊂ C. Hence K = C and M − A is the union of open sets, each with a degenerate boundary in A. Therefore, A is an A–set.

\underline{References}

1. F. B. Jones, “Aposyndetic Continua and Certain Boundary Problems,” American Journal of Mathematics, 63 (1941), 545–553.

2. G. T. Whyburn, Analytic Topology, 1st ed., American Mathematical Society Colloquium Publications 28, American Mathematical Society, Providence, RI, 1942.