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I am self-studying the book, Walter Rudin, Principles of Mathematical Analysis, and I am doing the Exercise 2.26.

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As mentioned in the hint, Exercise 23 suggests: every separable metric space has a countable base. According to the definitions, a metric space is separable if it contains a countable dense subset. A base $\{V_\alpha\}$ is a collection of sets such that every open set in X is the union of a subcollection in ${V_\alpha}$.

Exercise 24 suggests that, if in a metric space $X$, every infinite subset has a limit point, then $X$ is separable.

Following the hint, $X$ is separable, which means $X$ has a countable (infinite but countably many) base.

To prove $X$ is compact, we need to show for any open cover $\{G_\alpha\}$, there exists a finite subcover $\{G_1, G_2, \cdots, G_n\}$.

If I understand correctly, the hint and solution suggest first finding a countable subcover and then finding a finite subcover.

However, I do not see why "It follows that every open cover of $X$ has a countable subcover $\{G_n\}_{n=1}, n=1, 2, 3, \cdots$."

This was not proved in the hint or solution. Maybe trivial, but I can't see it at this moment. Can anyone give a hint? Thanks.

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Let $(U_i)_{i\in \mathbb{N}}$ a countabla basis, and consider an open covering $(V_j)_{j\in J}$, for every $x\in X$, there exists $j_x\in J$ such that $x\in V_{j_x}$. There exists $i_x\in \mathbb{N}$ such that $x\in U_{i_x}\subset V_{j_x}$ since $(U_i)_{i\in\mathbb{N}}$ is a basis of the topology. Consider the map $f:X\rightarrow\mathbb{N}$ defined by $f(x)=i_x$ and the map $g:\mathbb{{N}}\rightarrow J$ defined by $g(n)=j_y$ where we choose one $y\in X$ and $f(y)=n$. $X$ is covered by $V_{g(n)}$. To see this, let $x\in X$, suppose that $g(i_x)=j_y$, we have $g(y)=i_x$. This implies that $U_{i_x}=U_{i_y}$ and $x\in U_{i_x}=U_{i_y}\subset V_{j_y}=V_{g(n)}$.

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  • $\begingroup$ Thanks. This enables me to continue with the rest of the proof. $\endgroup$ – Mou Apr 21 '19 at 22:44
  • $\begingroup$ Hi Tsemo, I somehow lost the big picture of the proof. Why does it lead to a contradiction that z cannot be a limit point of {x_m}? Thank you. $\endgroup$ – Mou Apr 22 '19 at 1:38
  • $\begingroup$ By reading another post, I now understand, if the post is correct, that B(z) contains only finite elements of E, and thus it could not be a limit point. I was stuck at the provided solution because it says z is not a limit point of {x_m}. This is always true no matter if it means {x_m} for some m, or {x_m: m>=n}. Indeed, we should determine if z is a limit point of E (not {x_m} in either sense). $\endgroup$ – Mou Apr 22 '19 at 2:00

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