## Generating topologies, filters and uniformities

Last update: 23 July 2014

## Generating topologies, filters and uniformities

Let $X$ be a set. A filter ${ℱ}_{2}$ on $X$ is coarser than a filter ${ℱ}_{1}$ on $X$ if ${ℱ}_{2}\supseteq {ℱ}_{1}\text{.}$ A topology ${𝒯}_{2}$ on $X$ is coarser than a topology ${𝒯}_{1}$ on $X$ if ${𝒯}_{2}\supseteq {𝒯}_{1}\text{.}$ A uniformity ${𝒳}_{2}$ on $X$ is coarser than a uniformity ${𝒳}_{1}$ on $X$ if ${𝒳}_{2}\supseteq {𝒳}_{1}\text{.}$

Homework: Let $X$ be a set and let ${𝒯}_{1}$ and ${𝒯}_{2}$ be topologies on $X\text{.}$ Show that ${𝒯}_{2}\supseteq {𝒯}_{1}$ if and only if the identity map $idX: (X,𝒯1) ⟶ (X,𝒯2) x ⟼ x$ is continuous.

Homework: Let ${𝒳}_{1}$ and ${𝒳}_{2}$ be uniformities on $X\text{.}$ Show that ${𝒳}_{2}\supseteq {𝒳}_{1}$ if and only if the identity map $idX: (X,𝒳1) ⟶ (X,𝒳2) x ⟼ x$ is uniformly continuous.

Homework: Let $ℬ$ be a collection of subsets of $X\text{.}$ Let $𝒞=\left\{{B}_{1}\cap \cdots \cap {B}_{\ell } | \ell \in {ℤ}_{>0} \text{and} {B}_{1},\dots ,{B}_{\ell }\in ℬ\right\}\text{.}$ Show that $𝒯={⋃U∈𝒮U | 𝒮⊆𝒞}$ is a topology on $X$ and $𝒯$ is the minimal topology on $X$ containing .

Homework: Let $X$ be a set and let $C\subseteq X\text{.}$ Show that $𝒩(C)= {N⊆X | N⊇C}$ is a filter on $X\text{.}$

Homework: Let $ℱ$ be a filter on $X$ and let $ℬ\subseteq ℱ\text{.}$ Show that $ℱ$ is the minimal filter containing $ℬ$ if and only if $ℱ=⋃C∈𝒞 𝒩(C),$ where $𝒩\left(C\right)=\left\{N\subseteq X | N\supseteq C\right\},$ and $𝒞= { B1∩⋯∩Bℓ | ℓ∈ℤ>0 and B1,…,Bℓ∈B } .$

Homework: Let $𝒞$ be a collection of subsets of $X\text{.}$ Show that $ℱ=⋃C∈𝒞 𝒩(C),$ where $𝒩\left(C\right)=\left\{N\subseteq X | N\supseteq C\right\}$ is a filter on $X$ containing $𝒞$ if and only if $𝒞$ satisfies

 (a) $𝒞\ne \varnothing$ and $\varnothing \notin 𝒞,$ (b) if ${C}_{1},{C}_{2}\in 𝒞$ then there exists $C\in 𝒞$ such that $C\subseteq {C}_{1}\cap {C}_{2}\text{.}$

Homework: Let $ℬ$ be a collection of subsets of $X\text{.}$ Let $𝒞=\left\{{B}_{1}\cap \cdots \cap {B}_{\ell } | \ell \in {ℤ}_{>0} \text{and} {B}_{1},\dots ,{B}_{\ell }\in ℬ\right\}\text{.}$ Show that $ℱ=⋃C∈𝒞 𝒩(C),$ where $𝒩\left(C\right)=\left\{N\subseteq X | N\supseteq C\right\}$ is a filter on $X$ containing $𝒞$ if and only if $ℬ$ satisfies if $\ell \in {ℤ}_{>0}$ and ${B}_{1},{B}_{2},\dots ,{B}_{\ell }\in ℬ$ then ${B}_{1}\cap \cdots \cap {B}_{\ell }\ne \varnothing \text{.}$

Homework: Show that if $𝒳$ is a uniformity on $X$ then $𝒳$ is a filter on $X×X\text{.}$

Homework: Let $X$ be a set and let $𝒟$ be a collection of subsets of $X×X\text{.}$ Show that $𝒳=⋃D∈𝒟𝒱(D),$ where $𝒱\left(D\right)=\left\{V\subseteq X×X | V\supseteq D\right\}$ is a uniformity on $X$ containing $𝒟$ if and only if $𝒟$ satisfies

 (a) if ${D}_{1},{D}_{2}\in 𝒟$ then there exists $D\in 𝒟$ such that $D\subseteq {D}_{1}\cap {D}_{2},$ (b) if $D\in 𝒟$ then $\left\{\left(x,x\right) | x\in X\right\}\subseteq D\text{,}$ (c) if $D\in 𝒟$ then there exists $E\in 𝒟$ such that $E\subseteq \left\{\left(y,x\right) | \left(x,y\right)\in D\right\}\text{.}$ (d) if $D\in 𝒟$ then there exists $E\in 𝒟$ such that $E×XE= { (x,y) | there exists z∈X with (x,z)∈E and (z,y)∈E } ⊆D .$

Let $\left(X,𝒯\right)$ be a topological space and let $Y\subseteq X\text{.}$ The subspace topology on $Y$ is the minimal topology on $Y$ such that the inclusion $i: Y ⟶ X y ⟼ y$ is continuous.

Let $\left(X,𝒳\right)$ be a uniform space and let $Y\subseteq X\text{.}$ The subspace uniformity on $Y$ is the minimal uniformity on $Y$ such that the inclusion $i: Y ⟶ X y ⟼ y$ is uniformly continuous.

## Notes and References

The subspace topology, the product topology, and the minimal topology containing a collection of sets are discussed in [Bou, Ch I §2 No. 3 Examples]. The subspace uniformity and the product uniformity are defined in [Bou, Ch II §2 No. 4 Def. 3] and [Bou, Ch II §2 No. 6 Def 4]. The minimal filters containing a collection of sets are discussed in [Bou ChI §6 No. 2 Prop. 1 and Prop. 2].

These are a typed copy of handwritten notes from the pdf 140722GeneratingFilters.pdf.