, the free encyclopedia.
In topology and related fields of mathematics, a set U is called open if, intuitively speaking, you can "wiggle" or "change" any point x in U by a small amount in any direction and still be inside U. In other words, x can't be on the edge of U.
As a typical example, consider the open interval (0,1) consisting of all real numbers x with 0 < x < 1. If you "wiggle" such an x a little bit (but not too much), then the wiggled version will still be a number between 0 and 1. Therefore, the interval (0,1) is open. However, the interval (0,1] consisting of all numbers x with 0 < x ¡Ü 1 is not open; if you take x = 1 and wiggle a tiny bit in the positive direction, you will be outside of (0,1].
Note that whether a given set U is open depends on the surrounding space, the "wiggle room". For instance, the set of rational numbers between 0 and 1 (exclusive) is open in the rational numbers, but it is not open in the real numbers. Note also that "open" is not the opposite of "closed". First, there are sets which are both open and closed (called clopen sets);
in R and other connected spaces, only the empty set and the whole space are clopen
, while the set of all rational numbers smaller than ¡Ì2 is clopen in the rationals. Also, there are sets which are neither open nor closed, such as (0,1] in R.
Seem the compliment is open - ie. the null set, and hence it must be closed, but it's not because no points in it fail the test. --> "clopen
That answers that then :)
<< Fio >>