Open set (differential equation)

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In discussions about differential equations, the necessity of open sets, particularly in the context of defining functions like x' = f(t, x), is emphasized. The open set Ω is crucial because it ensures the continuity and differentiability of the function f, which are essential for the existence and uniqueness of solutions to the equations. Even if Ω is not open, it must include an open subset to maintain the meaningfulness of the differential equation. This requirement stems from the mathematical properties that govern the behavior of solutions in the vicinity of points within the domain. Understanding the role of open sets is fundamental to grasping the underlying principles of differential equations.
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Hello !

When I'm reading something about differential equations everywhere it's about open sets. For example when we define special kind of equation

x'=f(t,x)\,;\;f:\Omega\subset\mathbb{R}\times\mathbb{R}\to\mathbb{R}

Omega is open. Why Omega must be open? Thanks
 
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I don't know what is the full context. However even if Ω is not open, it must contain an open set for the diff. equ. to be meaningful.
 

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