Minima Nonblockers and Blocked Sets of a Continuum
Abstract
Given a continuum $X$ and an element $x \in X$, $\pi(x)$ is the smallest set that contains $x$ and does not block singletons, and $B(x)$ is the set of all elements blocked by ${x}$. We prove that for each $x \in X$, $B(x)$ is connected, $B(x) \subset \pi(x)$, and that if $B(x)$ is closed, then $B(x)=\pi(x)$. Among other results, we prove that if $X$ is a Kelley continuum and $\pi(x)$ is proper, then $B(x)=\pi(x)$. Finally, we prove that for a certain class of dendroids, the family of minima non-blockers coincides with the family of connected non-blockers.
- Publication:
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arXiv e-prints
- Pub Date:
- June 2023
- DOI:
- 10.48550/arXiv.2306.08897
- arXiv:
- arXiv:2306.08897
- Bibcode:
- 2023arXiv230608897P
- Keywords:
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- Mathematics - General Topology;
- 54B20 (Primary) 54F15
- E-Print:
- 13 pages, 3 figures, submitted to Topology and its Applications