# Thread: Compactness and Product Topologies

1. ## Compactness and Product Topologies

So, I have a problem that I'm working on and I can't seem to figure it out. We just started product topologies, so many properties are still new or unknown to me. Here's the problem:

Let $\displaystyle (X,\Omega)$ and $\displaystyle (Y,\Theta)$ be topological spaces. If $\displaystyle A \subseteq Y$ is compact relative to $\displaystyle \Theta$ and $\displaystyle x \in X$, show that $\displaystyle \{x\}\times Y$ is compact relative to the product topology on $\displaystyle X\times Y$.

I'm not seeing why $\displaystyle A$ being compact is sufficient for the whole product to be compact... Any help would be appreciated.

2. ## Re: Compactness and Product Topologies

Originally Posted by Aryth
Let $\displaystyle (X,\Omega)$ and $\displaystyle (Y,\Theta)$ be topological spaces. If $\displaystyle A \subseteq Y$ is compact relative to $\displaystyle \Theta$ and $\displaystyle x \in X$, show that $\displaystyle \{x\}\times Y$ is compact relative to the product topology on $\displaystyle X\times Y$.
I'm not seeing why $\displaystyle A$ being compact is sufficient for the whole product to be compact... Any help would be appreciated.
Are you sure that you have copied the question correctly?

Could it be show that $\displaystyle \{x\}\times A$ is compact relative to the product topology on $\displaystyle X\times Y~?$.

3. ## Re: Compactness and Product Topologies

Originally Posted by Plato
Are you sure that you have copied the question correctly?

Could it be show that $\displaystyle \{x\}\times A$ is compact relative to the product topology on $\displaystyle X\times Y~?$.
If it is then I certainly understand it, but the question definitely says $\displaystyle \{x\}\times Y$. I'll ask my professor tomorrow if the problem as stated has a typo and prove the revision instead. Thanks for the help.