Open Sets, Closed Sets and Limit Points
Rebuild openness, closure and limit points from balls alone, and restate continuity as preimages of open sets.
What a learner can do afterwards
- Prove that arbitrary unions and finite intersections of open sets are open
- Compute the closure and the interior of a given subset
- Prove that a function is continuous exactly when preimages of open sets are open
1 · Read
A set is open when every point brings a ball fully inside. Arbitrary unions of opens stay open, and finite intersections do too. Infinite intersections may collapse: nested intervals shrinking to a point leave a singleton with no room inside.
In the reals, the interior of the closed interval from 0 to 1 is the open interval, since the endpoints have no ball inside. The closure of the open interval adds exactly the two endpoints. The bare sequence 1 over n gains exactly its limit, so its closure adds one point.
Closed sets are complements of opens, and a set is closed exactly when it holds all its limit points. Closure adds every missing limit point, the smallest closed container. Interior shaves down to the largest open core, keeping only points with room to spare.
Continuity has a topological face: preimages of open sets are open. Pull back any open target and the source must be open too. A jump at zero betrays itself this way: the preimage of values near zero is a half line containing zero with no room to the right, which is not open.
Balls decide openness, limit points decide closedness, and preimages decide continuity.
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