# Thread: Closed and open set book example

1. ## Closed and open set book example

Consider the following subset of the real line:
$\displaystyle Y=[0,1]\cup (2,3)$,
in the subspace topology. In this space, the set [0,1] is open, since it is the intersection of an open set in R with Y. Similar (2,3) is open as a subset of Y; it is even open as a subset of R (I understand everything here). Since [0,1] and (2,3) are complements in Y of each other, we conclude that both [0,1] and (2,3) are closed as subsets of Y.

Y-[0,1] is open so [0,1] is closed.
Y-(2,3) is closed so why is (2,3) closed?

2. ## Re: Closed and open set book example

Originally Posted by dwsmith
Consider the following subset of the real line:
$\displaystyle Y=[0,1]\cup (2,3)$,
in the subspace topology. In this space, the set [0,1] is open, since it is the intersection of an open set in R with Y. Similar (2,3) is open as a subset of Y; it is even open as a subset of R (I understand everything here). Since [0,1] and (2,3) are complements in Y of each other, we conclude that both [0,1] and (2,3) are closed as subsets of Y.

Y-[0,1] is open so [0,1] is closed.
Y-(2,3) is closed so why is (2,3) closed?
The idea is this, let $\displaystyle [0,1]\subseteq Y$ is equal to $\displaystyle [0,1]\cap Y$ when $\displaystyle [0,1]$ is thought of as a subset of $\displaystyle \mathbb{R}$ and so closed in $\displaystyle Y$ with the subspace topology (being the intersection of $\displaystyle Y$ with a set closed in $\displaystyle \mathbb{R}$). Using the exact same idea you can show that $\displaystyle (2,3)$ is open in $\displaystyle Y$. But, let me ask you this, what is $\displaystyle Y\cap (-\infty,1.5)$?

3. ## Re: Closed and open set book example

Originally Posted by Drexel28
The idea is this, let $\displaystyle [0,1]\subseteq Y$ is equal to $\displaystyle [0,1]\cap Y$ when $\displaystyle [0,1]$ is thought of as a subset of $\displaystyle \mathbb{R}$ and so closed in $\displaystyle Y$ with the subspace topology (being the intersection of $\displaystyle Y$ with a set closed in $\displaystyle \mathbb{R}$). Using the exact same idea you can show that $\displaystyle (2,3)$ is open in $\displaystyle Y$. But, let me ask you this, what is $\displaystyle Y\cap (-\infty,1.5)$?
I understand why [0,1] is open and closed. I don't understand why (2,3) is closed though.

4. ## Re: Closed and open set book example

Originally Posted by dwsmith
I understand why [0,1] is open and closed. I don't understand why (2,3) is closed though.
Either because $\displaystyle (2,3)=Y-[0,1]$ or because $\displaystyle [2,3]=[1.5,10]\cap Y$.