---- Let $X$ be a [[poset|totally ordered set]]. > [!definition] Definition. ([[lower limit topology, upper limit topology]]) > The collection $\mathscr{B}^{-}$ of [[half-open interval|half-open intervals]] of the form[^1] $[a, b)=\{ x : a \leq x < b \}$ is a [[basis for a topology]] on $X$. We call the [[topological space|topology]] [[topology generated by a basis|generated]] by $\mathscr{B}^{-}$ the **lower-limit topology** on $X$ and denote it $\tau^{-}$. > The collection $\mathscr{B}^{+}$ of [[half-open interval|half-open intervals]] of the form $(a, b]=\{ x : a < x \leq b \}$ is likewise a [[basis for a topology]] on $X$. We call the [[topological space|topology]] [[topology generated by a basis|generated]] by $\mathscr{B}^{-}$ the **upper-limit topology** on $X$ and denote it $\tau^{+}$. > The [[intersection of topologies is a topology|meet]] $\tau:=\tau^{+} \cap \tau^{-}$ is precisely the [[order topology]] on $X$.[^2] [^1]: Here we are allowing $b=\infty$, i.e., we are including the [[ray|rays]] $[a, +\infty)$. [^2]: check this. ---- #### ---- #### References > [!backlink] > ```dataview > TABLE rows.file.link as "Further Reading" > FROM [[]] > FLATTEN file.tags as Tag > WHERE Tag = "#definition" OR Tag = "#theorem" OR Tag = "#MOC" OR Tag = "#proposition" OR Tag = "#axiom" > GROUP BY Tag > ``` > [!frontlink] > ```dataview > TABLE rows.file.link as "Further Reading" > FROM outgoing([[]]) > FLATTEN file.tags as Tag > WHERE Tag = "#definition" OR Tag = "#theorem" OR Tag = "#MOC" OR Tag = "#proposition" OR Tag = "#axiom" > GROUP BY Tag > ```