Inner points and breadth in certain compact semilattices

Abstract:

A point $x \in X$ is inner if there exists an open set $U$ containing $x$ such that for each open set $V$ with $x \in V \subseteq U$, the inclusion homomorphism ${i^* }:$: ${H^*}(X,X \setminus V) \to {H^*}(X,X \setminus U)$ is nontrivial. In this note it is proved that, if $X$ is a compact, chainwise connected topological semilattice of codimension $n$, and $x$ is a point of breadth $n + 1$, then $x$ is an inner point.