Can anyone tell me what open sets will look like for say $(0,1),[0,1),(0,1],[0,1]$ in the lower limit topology? I get what they will look like in the usual topology, but I am not sure what they will look like in the $\mathbb{R}_l$.
My guess would be, for $(0,1)_{l}$ open sets would look like $[a,b)$ where $a\ne 0,$ and $b\le 1$ that is $0<a<b\le1?$ Would appreciate the help.
Partitions and Equivalence Relations. A partition of a set $S$ is a family $P$ of pair-wise disjoint subsets of $S$ whose union is $S.$ An equivalence relation on $S$ is a binary relation $\sim$ on $S$ which is
(i) Reflexive: $\forall x\in S\;(x\sim x)$
(ii) Symmetric: $\forall x,y \in S\;(x\sim y\iff y\sim x) $
(iii) Transitive: $\forall x,y,z \in S\;((x\sim y \land y\sim z)\implies x\sim z.)$
Every partition on $S$ determines an equivalence relation on $S,$ and vice-versa: If $P$ is a partition of $S$ then for $x,y\in S$ let $x\sim y$ iff $x,y$ belong to the same member of $P.$ If $\sim$ is an equivalence relation on $S,$ then for each $x\in S$ let $[x]_{\sim}=\{y\in S: y\sim x\}.$ Then $P=\{[x]_{\sim}: x\in S\}$ is a partition of $S.$ (Note: $[x]_{\sim}$ is called an equivalence class.)
Notation: For $x,y \in \Bbb R$ let In$[x,y]=[x,y]\cup [y,x].$ That is, in the standard topology on $\Bbb R$ the set In$[x,y]$ is the closed interval from $x$ to $y$ (and vice-versa).
Let $S$ be open in the lower-limit topology. For $x,y\in S$ let $x\sim y \iff$ In$[x,y]\subset S.$ It is easily shown that
(i'). $\sim$ is an equivalence-relation on $S.$
(ii'). For $x\in S$ the equivalence-class $[x]_{\sim}=\{y\in S :y\sim x\}$ is a convex set of non-zero length.
(iii'). If $[x]_{\sim}$ is bounded above then it does not contain its sup, but if it is bounded below then it may or may not contain its inf .
The set $P$ of equivalence classes is a partition of $S$, so the equivalence-classes are pair-wise disjoint. Each equivalence class contains a rational, and the classes are pair-wise disjoint, so $P$ is finite or countably infinite.
And of course $S=\cup P.$
Observe that if $z=\sup \;[x]_{\sim}<\infty$ then $z\not \in S .$
Observe that if $z=\inf \;[x]_{\sim}\in S$ then $z\in [x]_{\sim}$ and $z=\sup\; (\; (-\infty, z)\setminus S\;).$
Examples. (1). Let $S=[0,1)$ or $S=(0,1).$ Then $P=\{S\}.$
(2).Let $C$ be the Cantor set. Let $[0,1]\setminus C =\cup \{(a_n,b_n):n\in \Bbb N\}$ where $(a_n,b_n)\cap (a_m,b_m)=\emptyset$ when $n\ne m.$ Let $S=\{[a_n,b_n): n\in \Bbb N\}.$ Each $[a_n,b_n)$ belongs to $P.$ Just as in the standard topology, open sets in the lower-limit topology can be "complicated".