Boundary Points Properties Quiz

Endpoints like 0 and 1 in (0,1) or [0,1] meet the definition of boundary points.

Only 0 touches both the interval and the outside (negative numbers). Points >0 do not touch anything outside.

Boundary points are always limit points or isolated points that still touch the complement; if no limit points exist, there is no boundary.

A boundary point must be in both closure(A) and closure(Aᶜ).

Being only in the closure of Aᶜ is not enough.

The points 0 and 1 are not in the interval, but every neighborhood of 0 or 1 touches the interval and the outside, so the boundary is {0,1}.

The endpoints 0 and 1 are where neighborhoods meet both the inside and outside of the interval, so the boundary is {0,1}.

Only the point 1 is a boundary point because intervals around 1 touch both (1,∞) and numbers less than 1.

A boundary point is one where every neighborhood touches both the set and its complement, meaning no matter how small the interval, it contains points inside A and outside A.

A boundary point may lie inside the set or outside the set, as long as every neighborhood intersects both A and its complement.

ℝ has no “outside,” so no point can touch both ℝ and its complement. The boundary is ∅.

Boundary points are always limit points, since they require neighborhoods that always meet the set.

Example: ℚ has empty interior but boundary ℝ.

Interior points require neighborhoods fully inside the set; boundary points require neighborhoods touching outside, so both cannot occur together.

This is not always true—taking the union can create new boundary points or remove some.

This fails in general because intersections may change which points lie on the edge.