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Let $X,Y$ be [[topological space]]s.
> [!proposition] Proposition. ([[local formulation of continuity]])
> The map $f:X \to Y$ is [[continuous]] if $X$ can be written as the union of [[open set|open sets]] $U_{\alpha}$ such that $f | U_{\alpha}$ is [[continuous]] for each $\alpha$.
> [!proof]- Proof. ([[local formulation of continuity]])
> Write $X=\bigcup_{\alpha}^{} U_{\alpha}$, where the function $f |U_{\alpha}$ is [[continuous]] for each $\alpha$. Let $V \subset Y$ be [[open set|open in]] $Y$. Then $f^{-1}(V) \cap U_{\alpha}= f^{-1}(V \cap f(U_{\alpha})) =(f |U_{\alpha})^{-1}(V),$
since each is "the set of points in $U_{\alpha}$ that are sent into $V$ by $f
quot;. This set is [[open set|open in]] $X$, since $f | U_{\alpha}$ is [[continuous]] for each $\alpha$. It follows that $f^{-1}(V) = \bigcup_{\alpha}^{}f^{-1}(V) \cap U_{\alpha}$
is also [[open set|open in]] $X$.
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#### 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
> ```