Continuum Without Non-Block Points

连续统的极小非阻塞集与被阻塞集

Minima Nonblockers and Blocked Sets of a Continuum

该拓扑学研究针对连续统上的非阻塞集与被阻塞集，证明了被阻塞集的连通性、包含关系等核心性质，推导了Kelley连续统等场景下两类集合相等的条件，并明确了一类树状连续统中极小非阻塞集与连通非阻塞集的等价性。

The Shore Point Existence Problem is Equivalent to the Non-Block Point Existence Problem

We prove the three propositions are equivalent: $(a)$ Every Hausdorff continuum has two or more shore points. $(b)$ Every Hausdorff continuum has two or more non-block points. $(c)$ Every Hausdorff continuum is coastal at each point. Thus it is consistent that all three properties fail. We also give the following characterisation of shore points: The point $p$ of the continuum $X$ is a shore point if and only if there is a net of subcontinua in $\{K \in C(X): K \subset κ(p) - p\}$ tending to $X$ in the Vietoris topology. This contrasts with the standard characterisation which only demands the …

Shore and Non-Block Points in Hausdorff Continua

We study the shore and non-block points of non-metric continua. We reduce the problem of showing a continuum to have non-block points to that of showing an indecomposable continuum to have non-block points. As a corollary we prove that separable continua have at least two non-block points -- and moreover are irreducible about their set of non-block points.

The hyperspace of non blockers of $F_1(X)$

A continuum is a compact connected metric space. A non-empty closed subset $B$ of a continuum $X$ does not block $x\in X\setminus B$ provided that the union of all subcontinua of $X$ containing $x$ and contained in $X\setminus B$ is dense in $X$. We denote the collection of all non-empty closed subset $B$ of $X$ such that $B$ does not block each element of $X\setminus B$ by $NB(F_1(X))$. In this paper we show some properties of the hyperspace $NB(F_1(X))$. Particularly, we prove that the simple closed curve is the unique continuum $X$ such that $NB(F_1(X))=F_1(X)$, given a positive answer to a…

The hyperspace of non-blockers of singletons, all the possible examples

Given a metric continuum $X$, a nonempty proper closed subspace $B$ of $X$, does not block a point $p\in X\setminus B$ provided that the union of all subcontinua of $X$ containing $p$ and contained in $X\setminus B$ is a dense subset of $X$. The collection of all nonempty proper closed subspaces $B$ of $X$ such that $B$ does not block any element of $X\setminus B$ is denoted by $NB(F_{1}(X))$. In this paper we prove that for each completely metrizable and separable space $Z$, there exists a continuum $X$ such that $Z$ is homeomorphic to $NB(F_{1}(X))$. This answers a series of questions by Cam…

A hereditarily indecomposable tree-like continuum without the fixed point property

A hereditarily indecomposable tree-like continuum without the fixed point property is constructed. The example answers a question of Knaster and Bellamy.