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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“No rules exist, and examples are simply life-savers answering the appeals of rules making vain attempts to exist.”
—André Breton (1896–1966)
“It is hardly to be believed how spiritual reflections when mixed with a little physics can hold people’s attention and give them a livelier idea of God than do the often ill-applied examples of his wrath.”
—G.C. (Georg Christoph)
“There are many examples of women that have excelled in learning, and even in war, but this is no reason we should bring ‘em all up to Latin and Greek or else military discipline, instead of needle-work and housewifry.”
—Bernard Mandeville (1670–1733)