Boundary (topology) - Examples

Examples

Consider the real line R with the usual topology (i.e. the topology whose basis sets are open intervals). One has

• ∂(0,5) = ∂ = ∂ = {0,5}
• ∂∅ = ∅
• ∂Q = R
• ∂(Q ∩ ) =

These last two examples illustrate the fact that the boundary of a dense set with empty interior is its closure.

In the space of rational numbers with the usual topology (the subspace topology of R), the boundary of, where a is irrational, is empty.

The boundary of a set is a topological notion and may change if one changes the topology. For example, given the usual topology on R2, the boundary of a closed disk Ω = {(x,y) | x2 + y2 ≤ 1} is the disk's surrounding circle: ∂Ω = {(x,y) | x2 + y2 = 1}. If the disk is viewed as a set in R3 with its own usual topology, i.e. Ω = {(x,y,0) | x2 + y2 ≤ 1}, then the boundary of the disk is the disk itself: ∂Ω = Ω. If the disk is viewed as its own topological space (with the subspace topology of R2), then the boundary of the disk is empty.