Interior, Boundary, and Closure

Real Analysis · Axiom Academy

Interior, Boundary, and Closure Three ways to look at a set on the number line — and the one tiny test that tells them apart: zoom in on a point and see what its neighborhood touches. One test sorts every point on the line Take a set S of real numbers and pick any point. There's a single question that decides everything about how that point sits relative to S : look at a tiny neighborhood around it — a short interval — and ask what it touches. If the whole neighborhood lies inside S , the point is interior . If every neighborhood, however small, hits both S and the outside, the point is on the boundary . If a neighborhood misses S entirely, the point is exterior . That one test builds the three operations this lesson is about. Watch a test point sweep across the half-open set S = (0, 1] . At each spot, a small neighborhood (the orange bracket) rides along, and the point is colored by what that neighborhood touches. As it goes, the picture of S — its interior in green, its two boundary points in teal — fills in beneath. The green stretch the sweep leaves behind is the interior (0,1) ; the two teal dots are the boundary . Drag a point, shrink its neighborhood

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