My goal is to prove that if given that the product space $X \times Y$ of two topological spaces is compact, then $X$ and $Y$ are compact. My idea for proof technique is as follows.
- Prove that the projection mappings $\operatorname{\pi}: X \times Y \to X$ and $\operatorname{\pi}: X \times Y \to Y$ are continuous.
- Prove that the image of the projection mapping is the whole space $X$ and $Y$.
- Suppose that either space $X$ or $Y$ is not compact. Use Theorem 26.5 from Munkres ("the image of a compact space under a continuous map is compact") to show that this results in a contradiction.
I am fairly confident in step one and step three is a gimme. My questioning lies in step 2. I am wondering:
- Is step 2 even true
- If it is true, is this something that is generally accepted in topology or if it needs proving.
I also am looking for feedback to see if as a whole this proof would be sound and valid. If anyone has alternate proof technique ideas, I would gladly be open to listening.
This community wiki solution is intended to clear the question from the unanswered queue.
If exactly one of the spaces $X , Y$ is empty, then the projection to the other space is not onto because $X \times Y = \emptyset$.
If $X, Y = \emptyset$ or $X, Y \ne \emptyset$, then 2. is true by definition of the product.