I have a presentation in which I am asked to demonstrate the following result:
A metric space $(X, d)$ is of the second category if and only if any sequence of non-empty open dense sets in $X$ has a non-empty intersection.
I have already completed the proof, using a bit of contradiction in both directions and some set algebra, along with definitions.
I would like my presentation to be more than just a simple demonstration; I want to explore implications (applications) directly from the result, any interesting facts, or anything that can enrich my presentation.
In my Mathematical Analysis III class, we are currently studying the Baire category theorem, which leads me to the following questions: Why is the Baire theorem important? What can we apply it to? What are spaces of the second and first category, intuitively speaking?
I hope u can help me