## Proper mappings

A morphism is a continuous function $f:X\to Y$.
A closed morphism is a morphism $f:X\to Y$ such that
 if $C$ is closed in $X$ then $f\left(C\right)$ is closed in $Y$.
A proper morphism is a continuous function $f:X\to Y$ such that
 if $Z$ is a topological space then $f×{\mathrm{id}}_{Z}:X×Z\to Y×Z$ is closed.

## Compact spaces

A topological space $X$ is quasicompact if the mapping $p:X\to \mathrm{pt}$ is proper.
A topological space is compact if it is quasicompact and Hausdorff.

Let $X$ be a set. An ultrafilter on $X$ is a maximal filter $ℱ$ (with respect to inclusion), i.e. a filter $ℱ$ such that there is no filter on $X$ which is strictly finer than $ℱ$.

Let $X$ be a topological space. The following are equivalent.

 (C) If $𝒥$ is a filter on $X$ then there exists $x\in X$ such that $x$ is a cluster point of $𝒥\text{.}$ (C') If $𝒢$ is an ultrafilter on $X$ then there exists $x\in X$ such that $x$ is a limit point of $𝒢\text{.}$ (C'') If $𝒞$ is a collection of closed sets such that $\bigcap _{K\in 𝒞}K=\varnothing$ then there exists $\ell \in {ℤ}_{>0}$ and ${K}_{1},\dots ,{K}_{\ell }\in 𝒞$ such that ${K}_{1}\cap {K}_{2}\cap \cdots \cap {K}_{\ell }=\varnothing \text{.}$ (C''') If $𝒮$ is a collection of open sets such that $\bigcup _{U\in 𝒮}U=X$ then there exists $\ell \in {ℤ}_{>0}$ and ${U}_{1},\dots ,{U}_{\ell }\in 𝒮$ such that ${U}_{1}\cup {U}_{2}\cup \cdots \cup {U}_{\ell }=X\text{.}$

 Sketch of proof. (C'') $⇔$ (C''') by taking complements. (C) $⇒$ (C'): Assume (C). To show: If $𝒢$ is an ultrafilter on $X$ then there exists $x\in X$ such that $x$ is a limit point of $𝒢\text{.}$ Assume $𝒢$ is an ultrafilter on $X\text{.}$ By (C), there exists $x\in X$ such that $x$ is a cluster point of $𝒢\text{.}$ Since $𝒢$ is an ultrafilter $x$ is a limit point of $𝒢\text{.}$ (C') $⇒$ (C): Assume (C'). To show: If $𝒥$ is a filter on $X$ then there exists $x\in X$ such that $x$ is a cluster point of $𝒥\text{.}$ Assume $𝒥$ is a filter on $X\text{.}$ Since the collection of filters on $X$ satisfies the hypotheses of Zorn's lemma, there exists an ultrafilter $𝒢$ such that $𝒢\supseteq 𝒥\text{.}$ By (C') there exists $x\in X$ such that $x$ is a limit point of $𝒢\text{.}$ So $x$ is a cluster point of $𝒢\text{.}$ Since $𝒢\supseteq 𝒥$ and $x$ is a cluster point of $𝒢$ then $x$ is a cluster point of $𝒥\text{.}$ (not C'') $⇒$ (not C): Assume that there is a collection $𝒞$ of closed sets such that $\bigcap _{K\in 𝒞}K=\varnothing$ and, if $\ell \in {ℤ}_{>0}$ and ${K}_{1},\dots ,{K}_{\ell }\in 𝒞$ then ${K}_{1}\cap \cdots \cap {K}_{\ell }\ne \varnothing \text{.}$ Let $𝒥$ be the set of subsets of $X$ which contain a set in $𝒞\text{.}$ Then $𝒥$ is a filter. Since $\bigcap _{N\in 𝒥}\stackrel{‾}{N}\subseteq \bigcap _{K\in 𝒞}\stackrel{‾}{K}=\bigcap _{K\in 𝒞}K=\varnothing ,$ $𝒥$ does not have a cluster point. (not C) $⇒$ (not C''): Assume that there exists a filter $𝒥$ on $X$ with no cluster point. Then $\bigcap _{N\in 𝒥}\stackrel{‾}{N}=\varnothing \text{.}$ Since $𝒥$ is a filter, if $\ell \in {ℤ}_{>0}$ and ${N}_{1},\dots ,{N}_{\ell }\in 𝒥$ then ${N}_{1}\cap \cdots \cap {N}_{\ell }\ne \varnothing$ and therefore $\stackrel{‾}{{N}_{1}}\cap \cdots \cap \stackrel{‾}{{N}_{\ell }}\ne \varnothing \text{.}$ Let $𝒞=\left\{\stackrel{‾}{N} | N\in 𝒥\right\}\text{.}$ Then $𝒞$ is a collection of closed sets such that $\bigcap _{K\in 𝒞}K=\varnothing$ but there does not exist ${K}_{1},\dots ,{K}_{\ell }\in 𝒞$ such that ${K}_{1}\cap {K}_{2}\cap \cdots \cap {K}_{\ell }=\varnothing \text{.}$ $\square$

Let $X$ be a Hausdorff topological space and let $K$ be a compact subset of $X$. Then $K$ is closed.

 Proof. Let $X$ be a Hausdorff topological space with topology $𝒯\text{.}$ Let $K\subseteq X\text{.}$ Assume $K$ is compact. To show: $K$ is closed. To show: ${K}^{c}$ is open. To show: If $x\in {K}^{c}$ then $x$ is an interior point of ${K}^{c}\text{.}$ Assume $x\in {K}^{c}\text{.}$ To show: There exists $U\in 𝒯$ such that $x\in U$ and $U\subseteq {K}^{c}\text{.}$ For each $y\in K$ there exist ${U}_{xy}\in 𝒯$ and ${V}_{xy}\in 𝒯$ such that $x\in {U}_{xy}$ and $y\in {V}_{xy}$ and ${U}_{xy}\cap {V}_{xy}=\varnothing \text{.}$ Then ${Vxy | y∈K}$ is an open cover of $X\text{.}$ Since $K$ is compact there exists $N\in {ℤ}_{>0}$ and ${y}_{1},{y}_{2},\dots ,{y}_{N}\in K$ such that ${ Vxy1, Vxy2, …, VxyN }$ is an open cover of $K\text{.}$ Let $U=Uxy1∩⋯∩ UxyN.$ Since $U$ is a finite intersection of open sets, $U$ is open. Also $x\in U$ and $U∩K = ( Uxy1∩⋯∩ UxyN ) ∩ ( Vxy1∪⋯∪ VxyN ) = ∅.$ So $x\in U$ and $U\subseteq {K}^{c}\text{.}$ So $x$ is an interior point of ${K}^{c}\text{.}$ So ${K}^{c}$ is open. So $K$ is closed. Let $x\in \stackrel{‾}{K}$. The neighbourhood filter $ℬ\left(x\right)$ of $x$ induces a filter ${ℬ}_{K}$ WHAT DOES THIS MEAN??? on $K$ which has a cluster point $y\in ????$. Since $ℬ\left(x\right)$ is coarser than ${ℬ}_{K}$ (considered as a filter base on $X$) the point $y$ is a cluster point of $ℬ\left(x\right)$. So $y=x$ since $X$ is Hausdorff. $\square$

Example-Homework: Let $X$ be a set with more than one point with topology $𝒯=\left\{\varnothing ,X\right\}\text{.}$ Show that every subset $A\subseteq X$ is compact, but the only closed subsets of $X$ are $\varnothing$ and $X\text{.}$ Note that $X$ is not Hausdorff.

## Locally ??? spaces

Let $E$ be a subset of $X$ and let $x$ be an element of ????. The set $E$ is locally closed at $x$ if there is a neighborhood ${N}_{x}$ of $x$ such that ${N}_{x}\cap E$ is closed in ${N}_{x}$.
The space $X$ is locally compact if each point of $x$ has a compact neighborhood.
The space $X$ is locally connected if each point of $X$ has a fundamental system of connected neighborhoods.

## Notes and References

This proof that compact implies closed in Hausdorff spaces is taken from notes of J. Hyam Rubinstein for a course Metric and Hilbert spaces at the University of Melbourne. The proof in [BR, Theorem 2.34] is of the same intent but stated only for metric spaces.

These notes follow Bourbaki [Bou, Ch.I §9 no. 1,2] and [Bou, Ch.I §10 no.1,2]. Theorem 2.1 characterizes compact spaces as spaces where limits exist. WHAT IS HEINE-BOREL??? WHAT IS BOREL-LEBESGUE??? The characterization of compact spaces by proper mappings is fundamental in algebraic geometry [REFERENCE?? EGA???].

Graphs of relations and functions are an interesting point. Maybe not. Put this in exercises? This seems to be special to compact and locally compact X and f: X to X/R, where R is an equivalence relation.

## References

[Bou] N. Bourbaki, General Topology, Springer-Verlag, 1989. MR1726779.

[BR] W. Rudin, Principles of mathematical analysis, Third edition, International Series in Pure and Applied Mathematics, McGraw-Hill 1976. MR0385023.

[Ru] W. Rudin, Real and complex analysis, Third edition, McGraw-Hill, 1987. MR0924157.