You are not logged in |
Barely alternating real almost chains and extension operators for compact lines
- Speaker(s)
- Maciej Korpalski
- Affiliation
- University of Wrocław
- Language of the talk
- English
- Date
- Oct. 9, 2024, 4:15 p.m.
- Room
- room 5050
- Seminar
- Topology and Set Theory Seminar
Consider an almost chain $\mathcal{A} = \{A_x \subset \omega: x \in X\}$ for some separable linearly ordered set $X$, so a family of subsets of $\omega$ such that for all $x < y, x, y\in X$ the set $A_x\setminus A_y$ is finite. Such a chain is barely alternating if for all $n \in \omega$ we cannot find elements $x_1 < x_2 < x_3 < x_4$ in $X$ satisfying $n \in A_{x_1}, A_{x_3}$, $n \notin A_{x_2}, A_{x_4}$.
We will show that under $MA(\kappa)$, if $|X| \leq \kappa$, then we can straighten our almost chain $\mathcal{A}$ into a barely alternating one by changing at most finitely many elements in each set $A_x$.
Next we will see how to use this fact to construct extension operator of small norm between spaces of continuous functions on a compact line and its countable discrete extension.
Joint work with Antonio Avil´es, [1] .
[1] Antonio Avil´es and Maciej Korpalski, Barely alternating real almost chains and extension
operators for compact lines, Rev. R. Acad. Cienc. Exactas F´ıs. Nat. Ser. A Mat. RACSAM
118 (2024), Paper No. 148.
