Relations among Equicontinuity, Characteristic 0 Property, Periodicity, and Their Variants

The paper’s Theorem A explicitly states that every equicontinuous semi-group action on a metrizable space is R-closed and of characteristic 0, and it proves this by: (i) defining R as the element-closure relation (y ∈ F(x) where F(x) denotes the closure of the element containing x), and D(x) as the bilateral prolongation via nets; (ii) showing characteristic 0 ⇔ R-closed for semi-decompositions on Hausdorff spaces (Lemma 3.1); and (iii) noting that equicontinuity of the action implies weak equicontinuity of the orbit semi-decomposition, which in turn yields characteristic 0 (Lemma 3.2), culminating in Theorem 3.3 and hence Theorem A (equicontinuous actions ⇒ R-closed and characteristic 0) . The candidate’s counterexample takes R to be the raw orbit relation {(x,y): y ∈ T(x)} and F_T(x) to be the orbit rather than the element closure, so their claim that R is not closed and F_T(x) ≠ D(x) is about a different relation and object than the paper’s R and F. Under the paper’s definitions (R is the orbit-closure relation; F(x) is the orbit closure), the example with T=(0,1] acting by scalar multiplication on ℝ does not contradict Theorem A: 0 belongs to the orbit-closure of every nonzero x, so (x,0) ∈ R; further, D(x)=F(x) in this example. The candidate’s proposed “correction” requiring compact T is therefore unnecessary and stems from the definitional mismatch, not from a flaw in the paper’s argument .