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.
Read more about this topic: Boundary (topology)
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