----- > [!proposition] Proposition. ([[smallest containing and largest contained topology]]) > Let $\{ \tau_{\alpha} \}$ be a family of of [[topological space|topologies]] on a set $X$. Then there exists a *unique* [[comparable topologies|smallest]] topology $\mathcal{T}_{\text{smallest}}$ on $X$ contain*ing* all the collections $\tau_{\alpha}$, and a *unique* [[comparable topologies|largest]] topology $\mathcal{T}_{\text{largest}}$ on $X$ contained *in* all $\tau_{\alpha}$. We call this the **topology generated by the $\tau_{\alpha}$**. > \ > Specifically, $\mathcal{T}_{\text{smallest}}$ may be obtained either by taking the intersection of all topologies containing all the collections $\tau_{\alpha}$, or as the [[topological space|topology]] generated by the [[subbasis for a topology|subbasis]] $\bigcup_{\alpha}^{} \tau_{\alpha}$. $\mathcal{T}_{\text{largest}}$ may be obtained as the intersection of the $\tau_{\alpha}$. ^b82d20 > [!proof]- Proof. ([[smallest containing and largest contained topology]]) > (a) Existence: Begin by defining $\mathcal{T}_{\text{smallest}}$ to be the [[intersection of topologies is a topology|intersection of all topologies]] containing all the collections $\{\tau_{\alpha}\}$; it follows that $\mathcal{T}_{\text{smallest}} \subset \mathcal{T}$ for all $\mathcal{T}$ containing all the collections $\{ \tau_{\alpha} \}$. Uniqueness: Suppose there is another $\mathcal{T}_{\text{smallest}}'$ with the property that $\mathcal{T}_{\text{smallest}}' \subset \mathcal{T}$ for all $\mathcal{T} \supset \bigcup_{\alpha}^{} \tau_{\alpha}$. Clearly $U \in \mathcal{T}_{\text{smallest}}'$ implies $U \in \bigcap_{}^{}\mathcal{T}$. Conversely, if $U \in \bigcap_{}^{}\mathcal{T}$ then $U \in \mathcal{T}_{\text{smallest}}'$ by the reasoning above. Therefore, $\mathcal{T}_{\text{smallest}}' = \bigcap_{}^{} \mathcal{T} = \mathcal{T}$. (a bit superfluous) \ As another approach, suppose that $\mathcal{T}_{\text{smallest}}$ is defined to be the [[topological space|topology]] generated by the [[subbasis for a topology|subbasis]] $\bigcup_{\alpha}^{}\tau _\alpha$. Let $\mathcal{T}$ be a [[topological space|topology]] on $X$ containing all the collections $\{ \tau_{\alpha} \}$. Then $\mathcal{T}_{\text{smallest}} \cap \mathcal{T}$ must contain all the collections $\{ \tau_{\alpha} \}$, implying that...? \ (b) Define $\mathcal{T}_{\text{largest}}$ as $\bigcap_{\alpha}^{}\tau_{\alpha}$. Suppose $\mathcal{T}$ is contained in all $\tau_{\alpha}$; $\mathcal{T} \subset \tau_{\alpha}$ for all $\alpha$. Then $\mathcal{T} \subset \bigcap_{\alpha}^{} \tau_{\alpha}$, hence $\mathcal{T} \subset \mathcal{T}_{\text{largest}}$. ^272a5e > [!basicexample] > If $X=\{ a,b,c \}$, let $\tau_{1} = \{ \emptyset, X, \{ a \}, \{ a,b \} \} \and \tau_{2}=\{ \emptyset,X,\{ a \}, \{ b,c \} \}.$ > Find the smallest [[topological space|topology]] containing $\tau_{1}$ and $\tau_{2}$ and the largest [[topological space|topology]] contained in $\tau_{1}$ and $\tau_{2}$. > \ Define $\mathcal{T}_{\text{smallest}}:=\{ \emptyset, X, \{ a \}, \{ a,b \}, \{ b, c \}, \{ a,b,c \}, \{ b \} \}$, and $\mathcal{T}_{\text{largest}}= \{ \emptyset, X, \{ a \} \}$ . ^725f1a ----- #### ---- #### 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 > ```