# Topology on set of "real lower bounds"

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}

A slight modification of this space, where you let $$t$$ range over $$[0,1]$$ instead of $$\mathbb R$$, is known as the double arrow space or the split interval. You can learn more about it here or here.

• Perfect—thanks for the pointers!
– Ziv
Commented Jun 24 at 7:52