questions on the completely accumulation Could somebody help me to understand the definition of completely accumulation? And help me show that this claim: A space $X$ is compact iff every infinite set in $X$ has a point of complete accumulation. Thanks ahead:)
 A: Let $X$ be a space and $A$ a subset of $X$. A point $x\in X$ is a point of complete accumulation of $A$ if $|U\cap A|=|A|$ for every open nbhd $U$ of $x$. Every point of the middle-thirds Cantor set $C$, for instance, is a point of complete accumulation of $C$: if $U$ is an open nbhd of a point of $C$, then $|U\cap C|=2^\omega=|C|$.
Suppose that $X$ is compact, and let $A\subseteq X$ be infinite. If $A$ does not have a point of complete accumulation, then each $x\in X$ has an open nbhd $U_x$ such that $|A\cap U_x|<|A|$. Let $\mathscr{V}$ be a finite subcover of $\{U_x:x\in X\}$, say $\mathscr{V}=\{V_1,\dots,V_n\}$. Then $A=\bigcup_{k=1}^n(A\cap V_k)$, so
$$|A|\le\sum_{k=1}^n|A\cap V_k|<|A|\;,$$
which is absurd. Thus, $A$ must have a point of complete accumulation.
Now suppose that $X$ is not compact, and let $\mathscr{U}=\{U_\xi:\xi<\kappa\}$ be an open cover of $X$ with no finite subcover. For $\eta<\kappa$ let $V_\eta=\bigcup_{\xi\le\eta}U_\xi$, and let $\mathscr{V}=\{V_\xi:\xi<\kappa\}$. By passing to a subset of $\mathscr{V}$ if necessary we may assume that $\kappa$ is a regular cardinal and that $V_\xi\subsetneqq V_{\xi+1}$ for all $\xi<\kappa$. For $\xi<\kappa$ let $x_\xi\in V_{\xi+1}\setminus V_\xi$, and let $A=\{x_\xi:\xi<\kappa\}$; then $A$ has no point of complete accumulation. To see this, let $x\in X$; then $x\in V_\eta$ for some $\eta<\kappa$, and $V_\eta$ is then an open nbhd of $x$ such that $$|A\cap V_\eta|=|\eta|<\kappa=|A|\;.$$
Added to explain the we may assume comment: Start with the original $\mathscr{V}=\{V_\xi:\xi<\kappa\}$. Recursively construct a strictly increasing subsequence $\langle\eta_\xi:\xi<\alpha\rangle$ of $\kappa$ such that $\{V_{\eta_\xi}:\xi<\alpha\}$ covers $X$ and $V_{\eta_\xi}\subsetneqq V_{\eta_{\xi+1}}$ for each $\xi<\alpha$. Let $\lambda$ be the cofinality of the ordinal $\alpha$, and let $\langle\gamma_\xi:\xi<\lambda\rangle$ be a cofinal subsequence of $\langle\eta_{\xi}:\xi<\alpha\rangle$. For $\xi<\lambda$ let $W_\xi=V_{\gamma_\xi}$; then $\mathscr{W}=\{W_\xi:\xi<\lambda\}$ is a strictly increasing open cover of $X$ of regular cardinality. Replace $\mathscr{V}$ by $\mathscr{W}$ if $\mathscr{V}$ did not already have these properties.
