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.

  • 1
    $\begingroup$ For $(0,1)_l$, the open set (in fact basic elements) are of the form $(0,b)$ for $0<b<1$ and $[a,b)$ for $0<a<b<1$ and $[a,1)$ also $\endgroup$ – Qurultay Mar 26 '18 at 13:14
  • $\begingroup$ $(0,1)=\bigcup_n[1/n,1)$, $[0,1)$ is open by definition. $[0,1]$ is closed, since it is the complement of the open set $\bigcup_n[-n,0)\cup[1-1/n,n)$. $\endgroup$ – user545497 Mar 26 '18 at 13:16
  • $\begingroup$ $(0,1]$ and $[0,1]$ cannot be open because every neighborhood of $1$ contains points outside of them. $(0,1]$ cannot be closed because $1/n\to 0\notin (0,1]$. $\endgroup$ – user545497 Mar 26 '18 at 13:22
  • $\begingroup$ Thank you for the help. Could anyone also help me with the remaining $3?$ For $[0,1)_l,$ will the open sets look like $(0,b),[a,b)$ for $0\le a <b\le 1?$ $\endgroup$ – Aurora Borealis Mar 26 '18 at 13:29
  • $\begingroup$ (0,b) is also open within lower limit (0,1). $\endgroup$ – William Elliot Mar 26 '18 at 21:20

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".

  • $\begingroup$ Ok this is alittle more than what I can absorb, I will try to understand this, thank you for the help. $\endgroup$ – Aurora Borealis Mar 27 '18 at 7:36
  • $\begingroup$ OK. Partitions and equivalence relations are a useful & widely used tool. $\endgroup$ – DanielWainfleet Mar 27 '18 at 18:48

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