This interactive page is intended to demonstrate the proof that if the complement \(X\backslash A\) of a set \(A\) is closed then \(A\) is open. Supposing that \(A\) is not open then there is a point \(a\in A\) with the property that every open ball \(B(a,\epsilon)\) has a point outside \(a\). Taking \(\epsilon\) to be \(1,1/2,1/3,1/4,\dots\) in turn, we generate a sequence, with all its terms in \(X\backslash A\), that converges to \(a\). This contradicts the closure of \(A\).
A yellow disc is displayed with a red dot at the centre that represents \(B(p,\epsilon)\) and the point \(p\)..
- Zoom in or out, by pressing the appropriate buttons.
- Press 'Next epsilon' to reduce the radius of the disc.
- Press `Next point in sequence` to choose a point inside \(A\cap B(p,\epsilon)\) .
- Press `Animate' to reduce \(\epsilon\) and plot the next point 50 times.
- Press 'Reset' to return to the initial state.