Non-cut Points Research Articles

Hereditarily decomposable continua have non-block points

In this note we expand upon our results from [1] to show that every nondegenerate hereditarily decomposable Hausdorff continuum has two or more non-block points, i.e. points whose complements contain a continuum-connected dense subset. The celebrated non-cut point existence theorem states that all nondegenerate Hausdorff continua have two or more non-cut points, and the corresponding result for non-block points is known to hold for metrizable continua. It is also known that there are consistent examples of Hausdorff continua with no non-block points, but that non-block point existence holds for Hausdorff continua that are either aposyndetic, irreducible, or separable.

Non-weak cut, shore and non-cut points in Whitney levels

We show that the property of having only non-weak cut points is a Whitney property, and that having only shore points is not a Whitney property. We also note that the property of having only non-cut (non-weak cut, shore) points is not a Whitney reversible property. This complements a result due to E. Matsuhashi.

On non-cut points and COTS

The concept of cut point convex sets is used to study non-cut points of a connected topological space. Some relations between cut point convex sets, H-sets and COTS are established. We prove that an H-set of a COTS is contained in a COTS with endpoints a and b for some a, b in the H-set. It is shown that if a connected topological space X has at most two non-cut points and an R(i) set that contains all the closed cut points of X, then X is a COTS with endpoints. Further we show that if every non-degenerate proper regular closed connected subset of a connected topological space X contains only finitely many closed points of X, then X has at least two non-cut points. A characterization of a non-indiscrete finite connected subspace of the Khalimsky line is also obtained.

On hyperspaces of non-cut sets of continua

In this paper we study the hyperspace of some kinds of non-cut sets of continua. This complements the study that has been done for non-cut points in continua by several authors.

Ultracoproduct continua and their regular subcontinua

We continue our study of ultracoproduct continua, focusing on the role played by the regular subcontinua—those subcontinua which are themselves ultracoproducts. Regular subcontinua help us in the analysis of intervals, composants, and noncut points of ultracoproduct continua. Also, by identifying two points when they are contained in the same regular subcontinua, we naturally generalize the partition of a standard subcontinuum of H⁎ into its layers.

Non-cut, shore and non-block points in continua

Non-cut, shore and non-block points in continua

Shore points of a continuum

In this paper we extend the study of shore points to every continuum and we prove that every continuum has at least two shore points. This result generalizes an important theorem in Continuum Theory: Every continuum has at leasts two non-cut points. We show that every point of irreducibility is a shore point, and we give some conditions under which a shore point is a point of irreducibility. Also, we characterize uniquely irreducible continua, an arc and a simple closed curve using shore points. Finally, we show that the union of shore points in a uniquely irreducible continua is a shore set.

A study of cut points in connected topological spaces

We prove that every H ( i ) subset H of a connected space X such that there is no proper connected subset of X containing H, contains at least two non-cut points of X. This is used to prove that a connected space X is a COTS with endpoints if and only if X has at most two non-cut points and has an H ( i ) subset H such that there is no proper connected subset of X containing H. Also we obtain some other characterizations of COTS with endpoints and some characterizations of the closed unit interval.

Cut points in some connected topological spaces

We prove that a connected topological space with endpoints has exactly two non-cut points and every cut point is a strong cut point; it follows that such a space is a COTS and the only two non-cut points turn out to be endpoints (in each of the two orders) of the COTS. A non-indiscrete connected topological space with exactly two non-cut points and having only finitely many closed points is proved homeomorphic to a finite subspace of the Khalimsky line. Further, it is shown, without assuming any separation axiom, that in a connected and locally connected topological space X, for a, b in X, S [ a , b ] is compact whenever it is closed. Using this result we show that an H ( i ) connected and locally connected topological space with exactly two non-cut points is a compact COTS with end points.

Formula omitted] connected topological spaces and cut points

A non-cut point existence theorem is proved for non-degenerate H ( i ) connected topological spaces. In an H ( i ) connected space, each component of the complement of a cut point contains a non-cut point. It is also shown that an H ( i ) connected topological space having exactly two non-cut points is a connected ordered topological space.

Shore points and noncut points in dendroids

A point p in a dendroid X is a shore point if there exists a sequence of subdendroids of X not containing p and converging to X with respect to the Hausdorff metric. The purpose of this paper is to present new results relating shore points and noncut points.

Hausdorff hypercubes which do not contain arcless continua

A Hausdorff arc is a compact connected Hausdorff space with exactly two noncut points. The finite product of a Hausdorff arc is called a Hausdorff hypercube. Suppose that X X is a Hausdorff arc which is first countable at none of its points and n n is a positive integer. We show that every nondegenerate subcontinuum of X n {X^n} contains a Hausdorff arc. Thus X n {X^n} contains no nondegenerate hereditarily indecomposable continuum.

The Orderability and Suborderability of Metrizable Spaces

A space is defined to be suborderable if it is embeddable in a (totally) orderable space. It is shown that a metrizable space X is suborderable iff (1) each component of X is orderable, (2) the set of cut points of each component of X is open, and (3) each closed subset of X which is a union of components has a base of clopen neighborhoods. Note that condition (1) and hence this result is topological since there are many good topological characterizations of connected orderable spaces. In a space X let Q denote the union of all nondegenerate components each of whose noncut points has no compact neighborhood. It is also shown that a metrizable space X is orderable iff (1) X is suborderable, (2) $X - Q$ is not a proper compact open subset of X, and (3) if W is a neighborhood of $p \in X$ and K is the component in X containing p such that $(W - K) - Q$ has compact closure and $\{ p\}$ is the intersection of the closures of $(W - K) - Q$ and $(W - K) \cap Q$, then K is a singleton. Corollaries are given; every condition in each of these corollaries is concisely stated and sufficient for a space to be orderable when it is metrizable and suborderable. Both of these results are extended to a class properly containing the metrizable spaces.

The orderability and suborderability of metrizable spaces

A space is defined to be suborderable if it is embeddable in a (totally) orderable space. It is shown that a metrizable space X is suborderable iff (1) each component of X is orderable, (2) the set of cut points of each component of X is open, and (3) each closed subset of X which is a union of components has a base of clopen neighborhoods. Note that condition (1) and hence this result is topological since there are many good topological characterizations of connected orderable spaces. In a space X let Q denote the union of all nondegenerate components each of whose noncut points has no compact neighborhood. It is also shown that a metrizable space X is orderable iff (1) X is suborderable, (2) X − Q X - Q is not a proper compact open subset of X, and (3) if W is a neighborhood of p ∈ X p \in X and K is the component in X containing p such that ( W − K ) − Q (W - K) - Q has compact closure and { p } \{ p\} is the intersection of the closures of ( W − K ) − Q (W - K) - Q and ( W − K ) ∩ Q (W - K) \cap Q , then K is a singleton. Corollaries are given; every condition in each of these corollaries is concisely stated and sufficient for a space to be orderable when it is metrizable and suborderable. Both of these results are extended to a class properly containing the metrizable spaces.

Dendritic compactifications of certain dendritic spaces

A dendritic space is a connected space in which every two points are separated by a third point. In this paper we describe a very natural method for obtaining a dendritic compactification of any connected space for which a dendritic compactification exists. The method is an extension of the familiar process of compactifying E1 by adjoining — oo and + oo. In what follows, an arc is a Hausdorff continuum with only two noncut points. A ray is an arc minus one of its noncut points. The space X is semi-locally connected at the point p if each open set containing p contains an open set V containing p such that X — V has at most finitely many components. LEMMA. If the space X is arcwise connected but is not semilocally connected at the point p, then there exists an open set U containing p such that if V is an open set containing p and lying in U, then X — V has infinitely many components that intersect both V and X- U. Proof. There exists an open set U containing p such that for each open set V containing p and lying in U, X — V has infinitely many components. Let V be an open set containing p and lying in U, and let S^ be the collection of all components of X — V that intersect both V and X — U. Suppose £f is finite. Let W be the union of V and all components of X — V lying in U. It follows from the arcwise connectivity of X that each component of X — V intersects V. Therefore W = X — (J Sf, so that W is open. But W^ U, X- W= Ό<9*,

Noncut points and modified compactness conditions

Noncut points and modified compactness conditions

On spaces without isolated noncut points

Introduction. Let X denote a topological space and f: X-*X a map. A point xEX is free if f(x) !x. f is called a +)-map if for every xeX, there exists a free point x' -x which does not separate f(x') and x. As a matter of convenience, we allow the identity map as a 4-map. Also, let F(f) denote the fixed points of f. If g: X-*X is another map, f and g are said to be F-homotopic provided there exists a homotopy H: X X I->X such that Ho =f, Hi = g and F(Ht) = F(f) for all t, 0<t<1. In [1], the author proves the following:

Topological semigroups and continua with cut points

In this paper we are concerned with continua that possess cut points and that admit the structure of topological semigroups, herein called mobs. First, we discuss the relations between the existence of cut points in compact connected mobs and the algebraic structure of the mob. Next, we consider D-chains, as defined by Remage [6], in compact connected mobs. D-chains may be considered to be generalizations of the classical A-sets. In the final section, we center our attention on compact metric trees, or dendrites, that are mobs and show that under certain circumstances the algebraic structure of the non-cut points has a marked influence upon the algebraic structure of the entire mob. The author wishes to acknowledge the advice and helpful suggestions of Professor A. D. Wallace in the preparation of this paper.