Open sets, closed sets and limit points on R

Try

The interval (0,2) is open because every point inside it has some small neighborhood still inside. At x=1/10, a radius smaller than 1/10 works; at x=1, a much larger radius works. Openness allows the radius to depend on the point. It does not require one radius that works simultaneously for every point, and it says nothing about whether a set has finite length.

Predict

Predict which of (0,2), [0,2], R and the empty set are open or closed. Closed means the complement in R is open, not merely that a drawing has solid endpoint dots. The whole line and empty set are both open and closed. A half-open interval is neither. Always identify the ambient space: a set open relative to a smaller domain need not be open in R.

Experiment

Use a neighborhood table.

Set	Test point	Why a neighborhood succeeds or fails
(0,2)	Any interior x	Radius below min(x,2−x) works
[0,2]	0	Every radius includes negative points
R without 0	Any x≠0	Radius |x|/2 avoids zero

Choose a point outside each set and test whether the complement is open there. This converts a geometric intuition into an explicit radius. Then examine the singleton containing zero: it is closed but no positive-radius neighborhood fits inside it.

Understand

An arbitrary union of open sets is open. A point in the union belongs to one member, and that member supplies a neighborhood contained in the union. A finite intersection is open because the minimum of finitely many positive radii is positive. Infinite intersections need not be open: intersecting (−1/n,1/n) over all positive integers gives only zero.

A closed set contains limits of convergent sequences of its points. If a purported limit lay outside, the open complement would contain a neighborhood of it, contradicting eventual membership of the sequence in that neighborhood. Conversely, if a set is not closed, some complement point has no neighborhood disjoint from the set. Choose a point of the set within 1/n of it for every n. This produces a convergent sequence whose limit is missing. Thus the sequential characterization follows in both directions.

Apply

A limit point of A is a point whose every neighborhood contains a point of A different from itself. A set is closed exactly when it contains all its limit points. Isolated points need not be limit points: a singleton is closed because it has no missing limit point, not because its sole member is approached by distinct members.

Repair “a closed set contains the limits of every sequence.” The sequence must consist of points of that set and actually converge. Closedness neither makes arbitrary sequences converge nor forces an outside constant sequence to enter the set. Track the quantified objects rather than shortening the statement until its restrictions vanish.

Continue

Optional depth studies closure and interior. The closure of A consists of points every neighborhood of which meets A; the interior consists of points having a neighborhood wholly inside A. For Q, density makes its closure R while irrational density makes its interior empty. These statements show that a set can be everywhere nearby without containing any interval.

Save a neighborhood proof and its sequential counterpart for the same closed set. The next lesson combines closedness with boundedness to obtain compactness, a stronger property that turns arbitrary open covers into finite ones and ensures subsequence limits remain in the domain.