# Thread: clopen sets in Std Topology

1. ## clopen sets in Std Topology

Which sets are closed and open in R^2 with the Std topologgy?

I am confused about this, help plz!! THis is a question that was made while looking at boundaries. Can you make a general statement?

2. Originally Posted by Andreamet
Which sets are closed and open in R^2 with the Std topologgy?

I am confused about this, help plz!! THis is a question that was made while looking at boundaries. Can you make a general statement?
The sets $\displaystyle \emptyset, \mathbb{R}^2$ are both open and closed.
Try it out, what is the definition of being open and closed?

3. Another example can be found in a disconnected topological space.

A topological space X is disconnected iff X has a proper subset A which is both open and closed.

For instance, the subspace $\displaystyle X=(0,1) \cup (2,3)$ of R is a disconnected space.
Both (0,1) and (2,3) are open in X.

Definition: A set C is closed in X if X\C is open.

Now, a complement of (0,1) in X is (2,3) which is open in X. Thus, (0,1) is a closed set in X. So, we can conclude (0,1) is a clopen set in X.

You can check that a boundary of (0,1) in X is empty as well (same with (2,3)).