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)
Famous quotes containing the word examples:
“Histories are more full of examples of the fidelity of dogs than of friends.”
—Alexander Pope (1688–1744)
“In the examples that I here bring in of what I have [read], heard, done or said, I have refrained from daring to alter even the smallest and most indifferent circumstances. My conscience falsifies not an iota; for my knowledge I cannot answer.”
—Michel de Montaigne (1533–1592)
“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)