> [!definition] > > Let $X$ be a [[Topological Space|topological space]]. A **compactification** of $X$ is a pair $(Y, e)$ such that > 1. $Y$ is a [[Compactness|compact]] [[Hausdorff Space|Hausdorff space]]. > 2. $e: X \to Y$ is an [[Topological Embedding|embedding]]. > 3. $e(E)$ is [[Dense|dense]] in $Y$.