On cohesive almost zero-dimensional spaces

Abstract

We investigate C-sets in almost zero-dimensional spaces, showing that closed σC-sets are C-sets. As corollaries, we prove that every rim-σ-compact almost zero-dimensional space is zerodimensional and that each cohesive almost zero-dimensional space is nowhere rational. To show these results are sharp, we construct a rim-discrete connected set with an explosion point. We also show that every cohesive almost zero-dimensional subspace of (Cantor set)×R is nowhere dense.

Dispersion points and rational curves

“…Added September 2019. Question 1 was recently answered by Jan J. Dijkstra and the author in [3]. Essentially, we use the Axiom of Choice to show ∇ .…”

Distinguishing endpoint sets from Erdős space

“…Observe also that since B is closed in and , the space E has a neighbourhood basis of C-sets of the form , where and W is clopen in Y . By [4, theorem 4·7] and the assumption that is connected, we see that is connected.…”

Distinguishing endpoint sets from Erdős space

“…Since B is closed in R × Y and Y ⊂ P, the space E is almost zero-dimensional, meaning that it has a neighborhood basis of closed sets which are intersections of clopen sets. The sets E ∩(−∞, t]×W , where t ∈ R and W is clopen in Y , form the required neighborhood basis for E. Observe also that [3,Theorem 4.7] and the assumption that Ẽ ∪ {∞} is connected to see that Ẽ \ (R × A) ∪ {∞} is connected. Supposing there is a space X and a homeomorphism h : Q × X → Ẽ, we will contradict this fact.…”