---- > [!theorem] Theorem. ([[for metrizable spaces, compact iff limit point compact iff sequentially compact]]) > Let $X$ be a [[metrizable]] [[topological space]]. Then the following are equivalent: > 1. $X$ is [[compact]]; > 2. $X$ is [[limit point compact]]; > 3. $X$ is [[sequentially compact]]. > [!proof]- Proof. ([[for metrizable spaces, compact iff limit point compact iff sequentially compact]]) > Denote by $d$ the [[metric]] [[metric topology|inducing the topology]] on $X$. > > That $(1) \implies (2)$ is true even when $X$ is not [[metrizable]]; this is already proven [[compactness implies limit point compactness, but not conversely|here]]. > > To show $(2) \implies (3)$, assume $X$ is [[limit point compact]]. > > Given a [[sequence]] $(x_{n})$ of points in $X$, consider the set $A:=\{ x_{n}: n \in \mathbb{N} \}$. If $A$ is finite, then $(x_{n})$ is eventually constant and so it is a [[converge|convergent]] [[subsequence]] of itself. So suppose $A$ is [[infinite]]. Then by [[limit point compact]]ness, $A$ has a limit point $x$. We define a [[subsequence]] of $(x_{n})$ converging to $x$ as follows: First choose $n_{1} \in \mathbb{N}$ so that $x_{n_{1}} \in B_{d}(x ,1).$ > Suppose $n_{i-1} \in \mathbb{N}$ is given. [[T1 axiom characterization of limit points|Because]] the ball $B_{d}\left( x, \frac{1}{i} \right)$ intersects $A$ in infinitely many points, we can choose an index $n_{i} > n_{i-1}$ s.t. $x_{n_{i}} \in B\left( x, \frac{1}{i} \right).$ > Then the [[subsequence]] $x_{n_{1}}, x_{n_{2}}, \dots$ [[converge]]s to $x$. > > Finally, we show $(3) \implies (1)$. From glance at Munkres this seems to require knowledge of the [[Lebesgue number lemma]] which we have not discussed yet. So it is [[TODO]]. > ---- #### ----- #### References > [!backlink] > ```dataview TABLE rows.file.link as "Further Reading" FROM [[]] FLATTEN file.tags GROUP BY file.tags as Tag > [!frontlink] > ```dataview > TABLE rows.file.link as "Further Reading" > FROM outgoing([[]]) > FLATTEN file.tags as Tag > WHERE Tag = "#definition" OR Tag = "#theorem" OR Tag = "#MOC" OR Tag = "#proposition" OR Tag = "#axiom" > GROUP BY Tag > ```