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I find the following two statements contradictory. Can anyone explain me why they are not?

  1. (from G. Gierz et al., Continuous Lattices and Domains, Proposition O-2.2) Let $L$ be a poset. For $L$ to be a complete lattice it is sufficient to assume the existence of arbitrary sups (or the existence of arbitrary infs).

  2. (from S. Willard, General Topology, 12B.1) The intersection of any number of filters on (a topological space) $X$ is a filter on X. But the set of all filters on $X$, ordered by $\mathcal{F}_1 \leq \mathcal{F}_2$ iff $\mathcal{F}_1 \subseteq \mathcal{F}_2$, is not a lattice because if $\mathcal{F}$ and $\mathcal{G}$ are different ultrafilters on $X$, then $\{\mathcal{F},\mathcal{G}\}$ has no supremum.

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  • $\begingroup$ The set of filters on $X$ has no arbitrary infs: there is no inf for the empty set of filters (this would be a maximum filter). $\endgroup$ – Henno Brandsma Sep 22 '15 at 18:35
  • $\begingroup$ Thank Henno for your prompt reply :) It has been very helpful. $\endgroup$ – Giorgio Bacci Sep 22 '15 at 19:31
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Try to carry out the dual of the argument in Gierz et al. on the poset $\mathfrak{F}$ of filters on $X$ ordered by $\subseteq$. For any $\mathfrak{X}\subseteq\mathfrak{F}$ this requires you to form $\mathfrak{B}=\bigcap\{\uparrow\!\!\mathscr{F}:\mathscr{F}\in\mathfrak{X}\}$, setting $\mathfrak{B}=\mathfrak{F}$ if $\mathfrak{X}=\varnothing$. Then you must take $\inf\mathfrak{B}$. When $\mathfrak{X}=\{\mathscr{F}_1,\mathscr{F}_2\}$, where $\mathscr{F}_1$ and $\mathscr{F}_2$ are ultrafilters, this requires you to take

$$\inf(\uparrow\!\!\mathscr{F}_1\cap\uparrow\!\!\mathscr{F}_2)=\inf\big(\{\mathscr{F}_1\}\cap\{\mathscr{F}_2\}\big)=\inf\varnothing\;.$$

But in any poset $\inf\varnothing$ is the maximum element of the poset if it exists and is otherwise undefined. $\mathfrak{F}$ has no maximum element, so $\inf\varnothing$ is undefined. Thus, $\langle\mathfrak{F},\subseteq\rangle$ does not have arbitrary infs after all: it only has infs of non-empty subsets.

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  • $\begingroup$ @Giorgio: You’re welcome. $\endgroup$ – Brian M. Scott Sep 22 '15 at 19:37
  • $\begingroup$ @BrianM.Scott thx for working out the longer version of my comment :) $\endgroup$ – Henno Brandsma Sep 22 '15 at 20:38

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