Examples
- In any space, the interior of the empty set is the empty set.
- In any space X, if, int(A) is contained in A.
- If X is the Euclidean space of real numbers, then int = (0, 1).
- If X is the Euclidean space, then the interior of the set of rational numbers is empty.
- If X is the complex plane, then int
- In any Euclidean space, the interior of any finite set is the empty set.
On the set of real numbers one can put other topologies rather than the standard one.
- If, where has the lower limit topology, then int = [0, 1).
- If one considers on the topology in which every set is open, then int = .
- If one considers on the topology in which the only open sets are the empty set and itself, then int is the empty set.
These examples show that the interior of a set depends upon the topology of the underlying space. The last two examples are special cases of the following.
- In any discrete space, since every set is open, every set is equal to its interior.
- In any indiscrete space X, since the only open sets are the empty set and X itself, we have int(X) = X and for every proper subset A of X, int(A) is the empty set.
Read more about this topic: Interior (topology)
Famous quotes containing the word examples:
“No rules exist, and examples are simply life-savers answering the appeals of rules making vain attempts to exist.”
—André Breton (18961966)
“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 (15331592)
“It is hardly to be believed how spiritual reflections when mixed with a little physics can hold peoples attention and give them a livelier idea of God than do the often ill-applied examples of his wrath.”
—G.C. (Georg Christoph)