---- > [!definition] Definition. ([[compactly generated]]) > A [[topological space]] $X$ is said to be **compactly generated** if it satisfies the following condition: $A \subset X$ is open in $X$ whenever $A \cap K$ is [[subspace topology|open in]] $K$ for each [[compact]] subspace $K \subset X$. ^definition > [!basicexample] > - [[locally compact]] $\implies$ compactly generated > - [[first-countable space|first-countable]] $\implies$ compactly generated ^basic-example > [!basicproperties] > - *(The fundamental property)* If $X$ is compactly generated, then a function $f:X\to Y$ is [[continuous]] if for each compact subspace $K$ of $X$, the restriction $f |_{K}$ is [[continuous]].[^1] > > > [!proof]- Proof. > > Suppose $f:X\to Y$ is such that its restrictions to compacts are all continuous. Let $V \subset Y$ be open. Then $f ^{-1}(V) \cap K= (f |_{K})^{-1}(V)$ > > is open in $K$. Since $X$ is compactly generated, $f ^{-1} (V)$ is open in $X$. > - [[homsets are closed in compactly generated topologies of compact convergence]] [^1]: This property is essentially the reason for formulating the definition. ---- #### ---- #### 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 > ```