2104.00894
On the triviality of flows in Alexandroff spaces
Pedro J. Chocano, Manuel A. Morón, Francisco R. Ruiz del Portal
correcthigh confidence
- Category
- Not specified
- Journal tier
- Note/Short/Other
- Processed
- Sep 28, 2025, 12:56 AM
- arXiv Links
- Abstract ↗PDF ↗
Audit review
The paper proves Theorem 2.6: every flow on an Alexandroff T0-space is trivial, via minimal open neighborhoods and order preservation of the time-t maps, concluding local fixedness and then global triviality by subdivision. The candidate solution uses the same core ingredients (specialization order, minimal neighborhoods, continuity ⇒ monotonicity) and pushes a slightly leaner argument to the same conclusion. The reasoning aligns step-for-step with the paper’s method and result, and no missing hypotheses are required. See Theorem 2.6 and its proof in the uploaded note .
Referee report (LaTeX)
\textbf{Recommendation:} no revision
\textbf{Journal Tier:} note/short/other
\textbf{Justification:}
This short note cleanly shows that continuous flows on Alexandroff T0-spaces are necessarily trivial. The argument is elementary, self-contained, and leverages the specialization order and minimal neighborhoods intrinsic to Alexandroff spaces. The result is of conceptual interest in topological dynamics on posets and in approximation schemes using Alexandroff spaces.