Started reading this paper by Das & Savas. A few definitions are new I need to have them clear. So here are those $4$ definitions and I've put down my own assessments. Kindly tell me if those are correct and if not what went wrong and what the right meaning would be. Thanks.
$1)$At one point it defines
Every Linear Topology $\tau$ on a vector space $L$ has a base $N$ for the neighbourhoods of $\theta$ satisfying the following properties:
a) Each $V ∈ N$ is a balanced set, that is $λx ∈ V$ holds for all $x ∈ V$ and every $λ ∈ \mathbb R$ with $|λ| ≤ 1$ . b) Each $V ∈ N$ is an absorbing set, that is for every $x ∈ L$, there exists a $λ > 0$ such that $λx ∈ V$ . c) For each $V ∈ N$ there exists some $W ∈ N$ with $W + W ⊂ V$.
Then next at Definition $2.2$ it says :
A Linear Topology $\tau$ on a Reisz Space $L$ is said to be lically solid if $\tau $ has a base at zero consisting of solid sets.
I think zero and $\theta$ are the same point for this linear topological space and the terms are interchangeable right?
The next things are the definitions $D_{\alpha},F_0,I_0.$
$2)$For a fixed $\alpha\in D; D_{\alpha}=\{\beta\in D:\beta\ge \alpha\}$
Then,is it valid to say that the set $D_{\alpha}$ is bounded below by $'\alpha'$?
$3)$ $$F_0=\{A\subset D:A\supset D_\alpha \text{ for some } \alpha\in D\}.$$ I'm guessing this $F_0$ is actually a collection of subsets of $D$ that are unbounded above and bounded below. That is, all the the subsets of $D$ that start from some $\alpha\in D$ and goes all the way up,are the members of $F_0.$ But then $D\supset D_\alpha$ is true for all $\alpha.$ So $D\in F_0 ?$
$4)$ $$I_0=\{A\subset D:A^c\in F_0\}\\=\{A\subset D: A^c\supset D_\alpha \text{ for some } \alpha\in D\}\\ \{A\subset D:A\subset D_\alpha^c\}\\=\{A\subset D: A=\{\gamma\in D:\gamma \le \alpha\}\text{ for some } \alpha\in D\}$$ So, is it safe to say $I_0$ is the set of all bounded above unbounded below subsets of $D?$
$5)$ Definition $2.3$: A net $\{s_α : α ∈ D\}$ in a topological space $(X, τ)$ is said to be $I$-convergent to $x_0 ∈ X$ if for any open set $U$ containing $x_0, \{α ∈ D : s_α < U\} ∈ I$.
Why using this strange notation $ s_α < U\ ?$ Apparently it means the same as $s_\alpha \notin U$ but what is the significance of writing it as $ s_α < U?$
These are from the second page of the paper from definitions $2.1,2.2$ and $2.3$
1) I'm pretty sure that by linear topological space, they mean topological vector space, and if so, then I suspect by 'base at zero', they mean that the topology has neighborhood basis at the zero vector.
2) $D_{\alpha}$ is the filter of $\alpha$ in $D$.
3) $F_0$ is the principal filter of $D_{\alpha}$ in $D$.
4) $I_0$ is principal ideal of $D_{\alpha}$ in $D$. (ideals are the dual notion to that of filters)
You may find this helpful: https://en.wikipedia.org/wiki/Filter_(mathematics)