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Definition: A topological space $X$ is said to be countably compact if every countable open covering of $X$ has a finite subcollection that also covers $X$.

Definition: A topological space $X$ is said to be limit-point-compact if every infinite subset of $X$ has a limit point in $X$.

Then how to prove the following result?

Let $X$ be a $T_1$-space. Then $X$ is countably compact if and only if $X$ is limit point compact.

My effort:

Suppose $X$ is countably compact. If $X$ is not limit point compact, then let $A$ be an infinite subset of $X$ such that $A$ has no limit point in $X$.

Let $$ B \colon= \left\{ b_1, b_2, b_3, \ldots \right\} $$ be a countably infinite subset of $A$.

Since we have assumed that $A$ has no limit point in $X$ and since $B \subset A$, therefore $B$ has no limit point in $X$ either. So, the set $B^\prime$ of all the limit points of $B$ in $X$ is empty and thus contained in $B$; hence $B$ is closed in $X$.

Since $B$ has no limit points in $X$ and since $B \subset X$, none of the elements of the set $B$ itself is a limit point of $B$; so for each element $b_n \in B$, there is an open set $U_n$ in $X$ such that $$ U_n \cap B = \left\{b_n \right\} \tag{1} .$$

Now the collection $$ \left\{ \ U_n \ \colon \ n \in \mathbb{N} \ \right\} \bigcup \{ X-B \} $$ forms a countable open covering of the countably compact space $X$, so some finite subcollection of this covering also covers $X$ and hence that finite subcollection also covers $B$. But the set $X-B$ contains no point of $B$. So $B$ is covered by finitely many of the sets $$ \left\{ \ U_n \ \colon \ n \in \mathbb{N} \ \right\}, $$ each of which contains exactly one point of set $B$ [Refer to (1) above.]. This implies that set $B$ is a finite set, contrary to our choice of $B$. Hence $X$ is limit point compact.

Is this proof correct?

How to prove the converse?

PS:

From the above proof (where we haven't required $X$ to be a T$_1$-space), we can even state the following:

Every (countably) compact topological space $X$ is also limit point compact.

Am I right?

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For the sake of contradiction, suppose that $X$ is limit-point compact but not countably compact. Then, there exists a countable open cover $(U_n)_{n\in\mathbb N}$ that admits no finite subcover. Choose $x_1\notin V_1\equiv U_1$. There exists some $n_1\in\mathbb N$ such that $x_1\in U_{n_1}$. Now choose $$x_2\notin V_2\equiv U_1\cup\ldots\cup U_{n_1}.$$ Choose $n_2\in\mathbb N$ such that $x_2\in U_{n_2}$. Then pick $$x_3\notin V_3\equiv U_1\cup\ldots\cup U_{n_1}\cup\ldots\cup U_{n_2}.$$ Pick $n_3\in\mathbb N$ such that $x_3\in U_{n_3}$. And so forth (note that $n_1<n_2<n_3<\ldots$). This way, one can construct an infinite set $D\equiv(x_1,x_2,\ldots)$ and an increasing sequence of open sets $(V_n)_{n\in\mathbb N}$ such that \begin{align*} x_m\notin V_n\quad\text{for any $m,n\in\mathbb N$ such that $m\geq n$}.\tag{$\clubsuit$} \end{align*} Note that $(V_n)_{n\in\mathbb N}$ is an open cover of $X$ as well. Also, the set $D$ infinite, since the points $x_1,x_2,\ldots$ are all distinct.

Now, if $X$ is limit-point compact, then the infinite set $D$ has a limit point $x\in X$. Let $m_0\in\mathbb N$ be such that $x\in V_{m_0}$ (remember that $(V_n)_{n\in\mathbb N}$ is an open cover). By the limit point property, there exists some $m_1\in\mathbb N$ such that $x_{m_1}\neq x$ and $x_{m_1}\in V_{m_0}$. One then must have $m_1<m_0$; see ($\clubsuit$). Since $X$ is $T_1$, the set $\{x_{m_1}\}^c$ is open and $$x\in V_{m_0}\cap\{x_{m_1}\}^c,$$ so there must exist some $m_2\in\mathbb N$ such that $x_{m_2}\in V_{m_0}\cap\{x_{m_1}\}^c$ (in particular, $x_{m_2}\neq x_{m_1}$, so $m_2\neq m_1$) and $x_{m_2}\neq x$. It follows again by ($\clubsuit$) that $m_2<m_0$. But then $$x\in V_{m_0}\cap \{x_{m_1}\}^c\cap\{x_{m_2}\}^c,$$ and so forth. Because of the limit-point property, one should be able to continue this pattern indefinitely. The problem is that one eventually runs out of distinct indices less than $m_0$, since the index set $\{1,\ldots,m_0-1\}$ is finite. Contradiction.


Your proof of the other direction seems correct to me.

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$\Rightarrow $ Suppose $A\subseteq X$ has no limit points. Wlog $A$ is countable so that $A=\left \{ a_{1}, a_{2},\cdots \right \}$.

Take $U_{n}=X-\left \{ a_{n},a_{n+1},\cdots \right \}$. Each $U_{n}$ is open in $X$ since, since $\left \{ a_{n},a_{n+1},\cdots , \right \}$ is a subset of $A$, and is therefore closed in $X$ because $A$ has no limit points. It is easy to see that $X=\bigcup _{n\geq 1}U_{n}$ so that $\left \{ U_{n} \right \}_{n\geq 1}$ is a countable open cover of $X$ which has no subcover.

$\Leftarrow $ Suppose that $X$ is not countably compact and let $\left \{ U_{n} \right \}_{n\geq 1}$ be a countable open cover of $X$ which has no subcover. Then we may choose, for each $n\in \mathbb N$ an $x_{n}\in X-\bigcup^{n} _{j=1}U_{j}$ such that $x_{i}\neq x_{k}$ if $i\neq k$. Set $A=\left \{ x_{n} \right \}_{n\in \mathbb N}$. Now choose $x\in X$. Then $x\in U_{L}$ for some $L\in \mathbb N$ since the $U_{n}$ form a cover of $X$. But by construction, $x_{i}\notin U_{L}$ as soon as $i\geq L$. Therefore, $U_{L}$ s a neighborhood of $x$ that intersects $A$ in only finitely many points and so $x$ is not a limit point of $A$. As $x$ was arbitrary, $A$ has no limit points in $X$.

Note: we used the fact that if $x$ is a limit point of a subset $A$ of a $T_{1}$ space $X$ then $A\cap (N_{x}-\left \{ x \right \})$ is infinite, for every neighborhood $N_{x}$ of $x$. For, if not, then there is an $N_{x}$ neighborhood of $x$ such that $A\cap (N_{x}-\left \{ x \right \})$ is finite and hence closed in $X$. But then, $U=N_{x}-(A\cap (N_{x}-\left \{ x \right \}))$ is open in $X$, contains $x$ and its intersection with $A$ is at most $\left \{ x \right \}$, which contradicts the fact that $x$ is a limit point of $A$.

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