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A space is normal if disjoint closed sets F, G have disjoint open neighborhoods. Every compact Hausdorff space and every metric space is normal.
You've studied the separation axioms — a hierarchy of conditions that capture how well a topological space can separate points and sets using open sets. Recall the key levels: T₁ (points are closed), T₂ or Hausdorff (disjoint points have disjoint open neighborhoods), and T₃ or regular (a point and a disjoint closed set have disjoint open neighborhoods). Normal spaces (T₄) push this one step further: *two disjoint closed sets* can always be separated by disjoint open neighborhoods.
To feel why this is a meaningful strengthening, compare T₃ and T₄. In a regular space, you can separate a *point* from a closed set it doesn't belong to. But a single point is a very special kind of closed set. Normality demands the same separation for *arbitrary* disjoint closed sets — even large, complicated ones. In some pathological spaces (certain infinite products, the long line), disjoint closed sets cannot always be separated this way, showing that normality is a genuine restriction.
The two major classes of normal spaces are metric spaces and compact Hausdorff spaces. For metric spaces, the proof is constructive: given disjoint closed sets F and G in a metric space, let U = {x : d(x,F) < d(x,G)} and V = {x : d(x,G) < d(x,F)}. These are open (they're defined by strict inequalities of continuous functions), disjoint, and contain F and G respectively. The metric provides enough structure to build the separating neighborhoods explicitly. For compact Hausdorff spaces, the argument is topological: compactness lets you build finite open covers, and Hausdorff-ness provides local separability that can be pieced together globally.
Why does normality matter? It is the exact condition needed for Urysohn's Lemma, which you'll study next: in a normal space, any two disjoint closed sets can be separated not just by open sets, but by a *continuous real-valued function* — one that equals 0 on one closed set and 1 on the other. This is a striking upgrade from separation by open sets alone, and it's the key to constructing continuous functions in topology. Urysohn's Lemma is then the engine behind the Tietze Extension Theorem, which says that every continuous real-valued function defined on a closed subset of a normal space extends to the whole space. Together, these results make normality the foundation of the theory of continuous functions on topological spaces.