----- > [!proposition] Proposition. ([[homsets are closed in compactly generated topologies of compact convergence]]) > Let $X$ be a [[compactly generated]] [[topological space|space]]; let $(Y,d)$ be a [[metric space]]. [[continuous|Then]] $C(X,Y):= \text{Hom}_{\mathsf{Top}}(X,Y)$ is [[closed set|closed]] in $Y^{X}$ in the [[topology of compact convergence]]. ^proposition > [!proposition] Corollary. > Invoking [[the sequence lemma]], we obtain the following. > Let $X$ be a [[compactly generated]] [[topological space|space]]; let $(Y,d)$ be a [[metric space]]. If a [[sequence]] of [[continuous]] functions $f_{n}:X \to Y$ [[converge|converges]] in to $f$ in the [[topology of compact convergence]], then $f$ is [[continuous]]. ^proposition > [!proof]- Proof. ([[homsets are closed in compactly generated topologies of compact convergence]]) > Let $f$ be a [[limit point]] of $C(X,Y)$; we wish to show $f$ is [[continuous]]. By the fundamental property of compactly generated spaces, it suffices to check $f |_{K}$ is [[continuous]] for each [[compact]] $K \subset X$. For each $n$, consider the basic open set[^1] $B_{K}\left( f , \frac{1}{n} \right)$; it must meet $C(X,Y)$ so let us choose $f_{n} \in C(X,Y) \cap B_{K}\left( f, \frac{1}{n} \right)$. Then $f_{n} |_{K} \to f |_{K}$ [[uniform convergence|uniformly]] on $K$, hence $f |_{K}$ is [[continuous]] by the [[uniform limit theorem]]. ----- #### [^1]: See [[topology of compact convergence]] to set notation. ---- #### 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 > ```