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For each ordinal 1 ≤ α < ω₁ we present separable metrizable spaces $X_{\alpha }$, $Y_{\alpha }$ and $Z_{\alpha }$ such that (i) $f{X}_{\alpha },f{Y}_{\alpha },f{Z}_{\alpha }=\omega ₀$, where f is either trdef or ₀-trsur, (ii) $A\left(\alpha \right)-trind{X}_{\alpha }=\infty$ and $M\left(\alpha \right)-trind{X}_{\alpha }=-1$, (iii) $A\left(\alpha \right)-trind{Y}_{\alpha }=-1$ and $M\left(\alpha \right)-trind{Y}_{\alpha }=\infty$, and (iv) $A\left(\alpha \right)-trind{Z}_{\alpha }=M\left(\alpha \right)-trind{Z}_{\alpha }=\infty$ and $A(\alpha +1)\cap M(\alpha +1)-trind{Z}_{\alpha }=-1$. We also show that there exists no separable metrizable space $W_{\alpha }$ with $A\left(\alpha \right)-trind{W}_{\alpha }\ne \infty$, $M\left(\alpha \right)-trind{W}_{\alpha }\ne \infty$ and $A\left(\alpha \right)\cap M\left(\alpha \right)-trind{W}_{\alpha }=\infty$, where A(α) (resp. M(α)) is the absolutely additive (resp. multiplicative) Borel class.

For a subspace A of a space X, a linear extender φ:C(A) → C(X) is called an $L_{ch}$-extender (resp. $L_{cch}$-extender) if φ(f)[X] is included in the convex hull (resp. closed convex hull) of f[A] for each f ∈ C(A). Consider the following conditions (i)-(vii) for a closed subset A of a GO-space X: (i) A is a retract of X; (ii) A is a retract of the union of A and all clopen convex components of X; (iii) there is a continuous $L_{ch}$-extender φ:C(A × Y) → C(X × Y), with respect to both the compact-open topology and...

On the set $ℝ$ of real numbers we consider a poset ${𝒫}_{\tau }\left(ℝ\right)$ (by inclusion) of topologies $\tau \left(A\right)$, where $A\subseteq ℝ$, such that $A_1\supseteq A_2$ iff $\tau \left(A_1\right)\subseteq \tau \left(A_2\right)$. The poset has the minimal element $\tau \left(ℝ\right)$, the Euclidean topology, and the maximal element $\tau \left(\varnothing \right)$, the Sorgenfrey topology. We are interested when two topologies ${\tau }_1$ and ${\tau }_2$ (especially, for ${\tau }_2=\tau \left(\varnothing \right)$) from the poset define homeomorphic spaces $(ℝ,{\tau }_1)$ and $(ℝ,{\tau }_2)$. In particular, we prove that for a closed subset $A$ of $ℝ$ the space $(ℝ,\tau (A\left)\right)$ is homeomorphic to the Sorgenfrey line $(ℝ,\tau (\varnothing \left)\right)$ iff $A$ is countable. We study also common properties...

Let $Zₙ={[0,1]}^{n+1}\setminus (0,1)ⁿ\times 0$. Then cmp Zₙ < def Zₙ for n ≥ 5. This is the answer to a question posed by de Groot and Nishiura [GN] for n ≥ 5.

We prove that $trt{K}_{\omega }>\omega +1$, where trt stands for the transfinite extension of Steinke’s separation dimension. This answers a question of Chatyrko and Hattori.