{ c} in Z, c is a constant integer.
(a,b) in R
[a,b] in R
{a/b} in Q, a/b is a constant rational number.
how can we judge whether these subsets are open ?
Can anyone give examples of open sets of these frequently asked set like R,Z,Q, C[0,1]?
{ c} in Z, c is a constant integer.
(a,b) in R
[a,b] in R
{a/b} in Q, a/b is a constant rational number.
how can we judge whether these subsets are open ?
Can anyone give examples of open sets of these frequently asked set like R,Z,Q, C[0,1]?
are we talking in the usual topology on the sets?
what does open mean? one definition says a set is open if and only if its compliment is closed. do you know what "closed" means? if not, use the definition of open directly. it is in your text. what can you come up with?
In analysis when you do some topology you define open in terms of a metric space. A set if defined to be open iff it contains its interior points and a set is defined closed iff it contains its boundary. There is an alternative definition. Openess is defined to be as above but closedness is defined in terms of a complement. There is a reason for that, eventhough this alternative definition is not as elegant. The reason is the notion of "topology". If you noticed metric spaces generalize the geometric notions that come up in Euclidean space. But we still need a metric. In topology we get rid of metric spaces altogether. Thereby making it even more general!
Here is the formal definition. Let $\displaystyle X$ be a set. Let $\displaystyle T\subseteq \mathcal{P}(X)$. We say $\displaystyle T$ is a topology over $\displaystyle X$ iff the following conditions are satisfied:
- If $\displaystyle \{ A_i | i\in I\}$ is a set of elements of $\displaystyle T$ then $\displaystyle \bigcup_{i\in I}A_i \in T$
- If $\displaystyle A,B \in T$ then $\displaystyle A\cap B \in T$
- $\displaystyle \emptyset, X \in T$
The elements of $\displaystyle T$ are called open subsets of $\displaystyle X$.
A subset $\displaystyle Y$ of $\displaystyle X$ (i.e. elements of $\displaystyle \mathcal{P}(X)$) is called closed iff $\displaystyle X - Y $ is open.
Notice that in the usual topology defined on $\displaystyle \mathbb{R}$ the open sets obey these properties. Any union of open sets is always open. But not intersection. Finite intersections of open sets is open, but it does not generalize to infinite intersections.