# ultrafilter and compact

*Let ( X,τ ) be a KC non-compact space. Then there is a discrete subset $D ⊆ X$, such that $\overline{D }$ is not compact. Furthermore there is an ultrafilter F in X, such that $D \in F$ and for every $C \in F$, $\overline{C }$ is not compact in $( X,τ )$.*

Proof. Let $U = \{U_{i} : i < \beta \}$ be a strictly increasing open cover of $X$, where$k$ is an infinite regular cardinal. We’ll construct sets $D_{\beta } = \{ x_{i }: i < \beta \}$ by transfini induction. First, let $D_{0} = \{x_{0} \}$ for some $x_{0 } ∈ U_{0}$. Let $\beta$ is ordinal successor. If $\overline{D}_{\beta - 1 }$ is compact, then there is $\alpha_{\beta}$ such that $\overline{D}_{\beta - 1 }$ ⊆ $U_{\alpha_{\beta}}$. Let $x_{ \beta } ∈ U_{\alpha+1} - U_{\alpha}$ and $D_{\beta } = D_{\beta - 1 } ∪ {x_{ \beta}}$. For limit ordinals $\beta$, let $D_{ \beta} = \bigcup_{i < \beta }D_{i}$. This process stops when $\overline{D}_{\beta}$ is not compact, which holds at least for $\beta = k$, because then the open cover $\mathcal{ U }$ witnesses that $\overline{D}_{k}$ is not compact. It is easy to see that $D_{ \beta}$ is discrete. The open set, which contains exactly one point $x_{i+1 }$ is $U_{ \alpha_{i+1}} - \overline{D_{i}}$. Let’s construct the ultrafilter. Let $F$ a filter-base, such that $D \in F$ and for any $C \in F$, $\overline {C }$ is not compact. Furthermore let $F$ be maximal with these properties. If$C \in F$ then for any $C^{′ } ⊇ C$, holds $C ^{′ }∈ F$, because if $C^{′}$ was compact, then it is closed from $KC$ property and $\overline{C} ⊆ C^{′}$, but $\overline{C }$ is non-compact closed subset of a compact set $C^{′}$, which is a contradiction. If $D = D_{0 } ∪ D_{1}$ and $D_{0} ∩ D_{1} = ∅$ then either $D_{0}$ or $D_{1}$ is in $F$. Otherwise both $\overline{D_{0 }}$ and $\overline{D_{1}}$ would be compact, hence $\overline{D} = \overline{D_{0 }} ∪ \overline{D_{1 }}$ would be compact. Finally this gives that $F$ is an ultrafilter.

(1): why we can say in the first paragraph" If $\overline{D}_{\beta - 1 }$ is compact, then there is $\alpha_{\beta}$ such that $\overline{D}_{\beta - 1 }$ ⊆ $U_{\alpha_{\beta}}$. Let $x_{ \beta } ∈ U_{\alpha+1} - U_{\alpha}$ and $D_{\beta } = D_{\beta - 1 } ∪ {x_{ \beta}}$"?

(2): why we can say " This process stops when $\overline{D}_{\beta}$ is not compact, which holds at least for $\beta = k$, because then the open cover $\mathcal{ U }$ witnesses that $\overline{D}_{k}$ is not compact. It is easy to see that $D_{ \beta}$ is discrete. The open set, which contains exactly one point $x_{i+1 }$ is $U_{ \alpha_{i+1}} -\overline{D_{i}}$."?

(3): what is the exact definition of infinite regular cardinal?

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The third question has no business in being here. You should also check out the Wikipedia page before asking it separately. –  Asaf Karagila Aug 4 '13 at 19:17
This is a duplicate of math.stackexchange.com/questions/464852/… –  Henno Brandsma Aug 11 '13 at 7:41