Specific question: Is there a name for the "topology of real lower bounds"? This is the order topology for the ordering $\supseteq$ on the set $$ \mathbb{LB} = \bigl\{ [t, \infty) \mid t \in \mathbb{R} \bigr\} \cup \bigl\{ (t, \infty) \mid t \in \mathbb{R} \bigr\}. $$ In addition to a name, is there a standard reference for its properties?
To clarify, I am not asking about a topology on $\mathbb{R}$ generated by $\mathbb{LB}$. I am asking about a topology on $\mathbb{LB}$ itself.
Motivation: I am studying a problem in operations research. The problem boils down to this: we open a box, find some amount of money $m$ inside, and have to accept or reject the money. I am specifically studying threshold policies for this problem, of which there are two types.
- The weak threshold $t$ policy, which I'll denote ${\geq}t$, is the policy that accepts the money if and only if $m \geq t$.
- The strict threshold $t$ policy, which I'll denote ${>}t$, is the policy that accepts the money if and only if $m > t$.
Formally, we can identify a threshold policy with the set of $m$ values it accepts, i.e. $$\begin{aligned} {\geq}t &= [t, \infty), \\ {>}t &= (t, \infty). \end{aligned}$$ So $\supseteq$ gives the "natural" ordering on threshold policies, e.g. ${\geq} 4$ is less than ${>}4$ is less than ${\geq}5$.
I'm studying a situation where I need to consider a limit of threshold policies. I cannot consider the thresholds $t$ alone, because taking a limit of policies can change the strictness of the threshold. For example, $$\begin{aligned} \lim_{\varepsilon \downarrow 0} {\geq}(t+\varepsilon) &= {>}t, \\ \lim_{\varepsilon \downarrow 0} {>}(t-\varepsilon) &= {\geq}t. \end{aligned}$$
