Open subsets of R^2

smiles

Open subsets right?

For R^2 (real numbers in the form (x,y) basically the plane)

A subset A of R^2 is called open in R^2 if every point of A has a radius of delta > 0 which is still contained within A.

eg x^2 + y^2 <= 1 is not open
but x^2 + y^2 < 1 is open.

Question: Is R^2 an open subset of R^2?

heard arguments for both sides, anyone definite?

<< Fio >>

Syth

A point is an element of the boundary of a set if, for all r > 0, the circle centred at that point contains points in the original set and point not in the set.

Thus R² doesn't have a boundary as there is no point on the plane in which every circle centred at the point contains points not on the plane.

A set is closed if the set includes all of it's boundary, and open if it contains none of it's boundary. It is neither open or closed if it contains some (but not all) of it's boundary.

Since R² doesn't have a boundary, it can't be closed. (by this definition).

By your definition, R² is also open. The 2 definitions seem to be identical.

smiles

clopen

From Wikipedia, 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 >>

Syth

Yep my calculus lecturer tryieds to catch us with this. R² has no boundary points, so the set of boundary points is the null set. R² contains the null set, so it contains all it's bonudary set, so it's closed, but it has no boundary points, so it's open. Open and closed.