----- > [!proposition] Proposition. ([[compactness implies limit point compactness, but not conversely]]) > If a [[topological space]] $X$ is [[compact]], then $X$ is [[limit point compact]]. The converse is, in general, false. > [!proof]- Proof. ([[compactness implies limit point compactness, but not conversely]]) > Suppose $X$ is [[compact]]. Take a subset $A$ of $X$ and suppose $A$ has no [[limit point]] in $X$. We wish to prove $A$ is finite. > Since $A$ trivially contains all its [[closure is set together with limit points|closed iff contains all limit points|limit points]] $A$ is [[closed set|closed]] in $X$. Around each $a \in A$ we can find $U_{a}$ s.t. $U_{a} \cap A=\{ a \}$. By [[compact|compactness of]] $X$, the [[cover|open cover]] $(X-A) \cup \{ U_{a} \}_{a \in A}$ has a finite subcover. Since each $U_{a}$ contains exactly one element of $A$, and it takes finitely many $U_{a}$ to cover $A$, we conclude that $A$ is finite. > Now we disprove the converse. Let $Y=\{ a,b \}$ be endowed with the [[discrete topology|trivial topology]]. Then the space $X:=\mathbb{N} \times Y$ > is [[limit point compact]], for *every* nonempty subset of $X$ has a [[limit point]]. It is not [[compact]], though, since the [[cover|covering]] of $X$ by the open sets $U_{n}=\{ n \} \times Y$ has no finite subcollection covering $X$. ----- #### ---- #### References > [!backlink] > ```dataview > TABLE rows.file.link as "Further Reading" > FROM [[]] > FLATTEN file.tags as Tag > WHERE Tag = "#definition" OR Tag = "#theorem" OR Tag = "#MOC" OR Tag = "#proposition" OR Tag = "#axiom" > GROUP BY 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 > ```