sets

can someone explain to me how a set can be both open and closed.

The greektard tried and failed at explaining it to me.

plus can you give me an example of a set that's neither open nor closed.

would this be something like (0,1] ??

Reply 1

generalebriety

What are your definitions of "open" and "closed"?

Reply 2

john !!

They are standard definitions. Open means that within a neighbourhood of any point in the set there exist only other points which are also in the set. Closed means that in a neighbourhood of a point in the set there are some points ot in the set.

For example, an open set is {x | 1 < x < 3} because there is always a larger number than any number you chose within the set which is also in the set (2.99, 2.999,.. etc.). A closed set is {x | 1 _< x _< 3} because if you take the point 3 in the set , the point 3+epsilon is in the neighbourhood of 3 but not in the set.

"open" and "closed" can apply to individual points, and the whole set is only open or closed if all points within it are open or closed respectively. So for example the set you give (0,1] would not be open or closed, I don't think.

Reply 3

Simplicity

Its called clopen. Look on wikipedia.

Reply 4

DFranklin

john !!

They are standard definitions. Open means that within a neighbourhood of any point in the set there exist only other points which are also in the set. Closed means that in a neighbourhood of a point in the set there are some points ot in the set.You might want to check those standard definitions. Your definition of an open set is imprecise (does 'a neighbourhood' mean any neighbourhood, or simply that we can find one particular neighbourhood?). Your definition of a closed set is wrong.

A closed set is simply one whose complement is open. In particular,

\mathbb{R}

is closed, contrary to your definition.

open" and "closed" can apply to individual points, and the whole set is only open or closed if all points within it are open or closed respectively.

This doesn't really make any sense. Singleton points are always closed under the standard topology.

Reply 5

Drederick Tatum

Since open sets are part of the definition of a topological space, it is easy to come up with sets that are open and closed, neither open or closed, open but not closed and closed but not open (if we can choose the topological space).

For example, the set X={1,2,3} with the collection T={{},X,{1},{1,2}} is a topological space with the set X open and closed, {1} open but not closed, {3} closed but not open and {1,3} neither open or closed.

Reply 6

Totally Tom

me thinks i'm gonna have to read a bit more on topological spacez.

cheers guiz.

Reply 7

DFranklin

Once you do the reading, you'll find there are two sets that (by definition) are always both open and closed in any(*) topological space.

Reply 8

generalebriety

john !!

They are standard definitions.

Well, there are several definitions, and I didn't want to work from one that he wasn't familiar with. (And as DFranklin pointed out, yours are wrong.)