Generating topologies, filters and uniformities

Arun Ram
Department of Mathematics and Statistics
University of Melbourne
Parkville, VIC 3010 Australia
aram@unimelb.edu.au

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 $$\begin{array}{cccc}{\text{id}}_X:& (X,{𝒯}_1)& ⟶& (X,{𝒯}_2)\\ & x& ⟼& x\end{array}$$ 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 $$\begin{array}{cccc}{\text{id}}_X:& (X,{𝒳}_1)& ⟶& (X,{𝒳}_2)\\ & x& ⟼& x\end{array}$$ is uniformly continuous.

Homework: Let $ℬ$ be a collection of subsets of $X\text{.}$ Let $𝒞=\{B_1\cap \cdots \cap B_{\ell }\hspace{0.17em}|\hspace{0.17em}\ell \in {\mathbb{Z}}_{>0}\hspace{0.17em}\text{and}\hspace{0.17em}{B}_1,\dots ,B_{\ell }\in ℬ\}\text{.}$ Show that $$𝒯=\left\{\bigcup _{U\in 𝒮}U\hspace{0.17em}\right|\hspace{0.17em}𝒮\subseteq 𝒞\}$$ is a topology on $X$ and $𝒯$ is the minimal topology on $X$ containing ℬ.

Homework: Let $X$ be a set and let $C\subseteq X\text{.}$ Show that $$𝒩\left(C\right)=\{N\subseteq X\hspace{0.17em}|\hspace{0.17em}N\supseteq C\}$$ 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 $$ℱ=\bigcup _{C\in 𝒞}𝒩\left(C\right),$$ where $𝒩\left(C\right)=\{N\subseteq X\hspace{0.17em}|\hspace{0.17em}N\supseteq C\},$ and $$𝒞=\{B_1\cap \cdots \cap B_{\ell }\hspace{0.17em}|\hspace{0.17em}\ell \in {\mathbb{Z}}_{>0}\hspace{0.17em}\text{and}\hspace{0.17em}{B}_1,\dots ,B_{\ell }\in B\}\text{.}$$

Homework: Let $𝒞$ be a collection of subsets of $X\text{.}$ Show that $$ℱ=\bigcup _{C\in 𝒞}𝒩\left(C\right),$$ where $𝒩\left(C\right)=\{N\subseteq X\hspace{0.17em}|\hspace{0.17em}N\supseteq C\}$ 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 $𝒞=\{B_1\cap \cdots \cap B_{\ell }\hspace{0.17em}|\hspace{0.17em}\ell \in {\mathbb{Z}}_{>0}\hspace{0.17em}\text{and}\hspace{0.17em}{B}_1,\dots ,B_{\ell }\in ℬ\}\text{.}$ Show that $$ℱ=\bigcup _{C\in 𝒞}𝒩\left(C\right),$$ where $𝒩\left(C\right)=\{N\subseteq X\hspace{0.17em}|\hspace{0.17em}N\supseteq C\}$ is a filter on $X$ containing $𝒞$ if and only if $ℬ$ satisfies if $\ell \in {\mathbb{Z}}_{>0}$ 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 $X\times X\text{.}$

Homework: Let $X$ be a set and let $𝒟$ be a collection of subsets of $X\times X\text{.}$ Show that $$𝒳=\bigcup _{D\in 𝒟}𝒱\left(D\right),$$ where $𝒱\left(D\right)=\{V\subseteq X\times X\hspace{0.17em}|\hspace{0.17em}V\supseteq D\}$ 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 $\left\{(x,x)\hspace{0.17em}\right|\hspace{0.17em}x\in X\}\subseteq D\text{,}$
(c)	if $D\in 𝒟$ then there exists $E\in 𝒟$ such that $E\subseteq \left\{(y,x)\hspace{0.17em}\right|\hspace{0.17em}(x,y)\in D\}\text{.}$
(d)	if $D\in 𝒟$ then there exists $E\in 𝒟$ such that $$E{\times }_{X}E=\left\{(x,y)\hspace{0.17em}\right|\hspace{0.17em}\genfrac{}{}{0ex}{}{\text{there exists}\hspace{0.17em}z\in X\hspace{0.17em}\text{with}}{(x,z)\in E\hspace{0.17em}\text{and}\hspace{0.17em}(z,y)\in E}\}\subseteq D\text{.}$$

Let $(X,𝒯)$ 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 $$\begin{array}{cccc}i:& Y& ⟶& X\\ & y& ⟼& y\end{array}$$ is continuous.

Let $(X,𝒳)$ 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 $$\begin{array}{cccc}i:& Y& ⟶& X\\ & y& ⟼& y\end{array}$$ 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.

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