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.