# Thread: f: X\to Y (Topology)

1. ## f: X\to Y (Topology)

X and Y are topological spaces. $\displaystyle f:X\to Y$ is said to be open if for every open set $\displaystyle U$ of $\displaystyle X$, the set $\displaystyle f(U)$ is an open set of Y. Show projections $\displaystyle \pi_1=X\times Y\to X$ and $\displaystyle \pi_2=X\times Y\to Y$ are open maps.

I have no idea what I need to do.

2. ## ~

Originally Posted by dwsmith
X and Y are topological spaces. $\displaystyle f:X\to Y$ is said to be open if for every open set $\displaystyle U$ of $\displaystyle X$, the set $\displaystyle f(U)$ is an open set of Y. Show projections $\displaystyle \pi_1=X\times Y\to X$ and $\displaystyle \pi_2=X\times Y\to Y$ are open maps.
As I said before, topology is all about definitions.
What is an open set in $\displaystyle X\times Y~?$. How is it constructed?
Call it $\displaystyle \mathcal{O}$.
What is the meaning of $\displaystyle \pi_1(\mathcal{O})$, i.e. what set is that?

3. ## Re: ~

Mathematics itself is largely "all about definitions". I once had a student ask about a problem in the text (Linear Algebra). I read the problem out to the class, wrote a single word from the problem on the chalk board, and asked "what is the definition of that word". I got a lot of blank stares so I sat down at my desk, placed my hands on the desk, and waited. Eventually, the students started leafing through the book and finally one even looked up the word in the index! Once they had the precise definition of the word, they could do the problem.

Definitions in mathematics are "working definitions"- you use the precise words of the definition in proofs and problems.

4. ## Re: ~

Originally Posted by Plato
As I said before, topology is all about definitions.
What is an open set in $\displaystyle X\times Y~?$. How is it constructed?
Call it $\displaystyle \mathcal{O}$.
What is the meaning of $\displaystyle \pi_1(\mathcal{O})$, i.e. what set is that?
I know that $\displaystyle X\times Y$ is the product top that is $\displaystyle U\times V$ where U and V are open sets in X and Y, respectively. The construction is the intersection of the open sets which is another open set.

5. ## Re: ~

Originally Posted by dwsmith
I know that $\displaystyle X\times Y$ is the product top that is $\displaystyle U\times V$ where U and V are open sets in X and Y, respectively. The construction is the intersection of the open sets which is another open set.
Well that is rather crude.
OK what do you think $\displaystyle \pi_1(U\times V)=~?$

6. ## Re: ~

Originally Posted by Plato
Well that is rather crude.
OK what do you think $\displaystyle \pi_1(U\times V)=~?$
That would be U then.

7. ## Re: ~

Originally Posted by dwsmith
I know that $\displaystyle X\times Y$ is the product top that is $\displaystyle U\times V$ where U and V are open sets in X and Y, respectively. The construction is the intersection of the open sets which is another open set.
As Plato said, that is rather crudely put. A "basis" for the product topology is the collection of all sets of the form $\displaystyle U\times V$ where U is in X and V is in Y. But then we must be able to take finite intersections and unions which can give open sets that are NOT of that form. For example, the open disk, $\displaystyle \{(x, y)| x^2+ y^2< 1\}$ is open in $\displaystyle R^2$ even though it is not of the form "$\displaystyle U\times V$" for any open U and V in R (which would be an open rectangle).