I'm an undergraduate pure mathematics major with relatively no foundation in pure mathematics. How would I go about filling the gaps?

Question

Here's the situation. I completed an associate's degree at a community college while in high school, and am now in my second (and final) year in an undergraduate pure mathematics program. While I have made it here by the skin of my teeth, I find myself at a place in my mathematical understanding wherein I am incapable of both thoroughly understanding a concept, as well as unable to know how/where to apply a topic when I do feel as though I get it.

My questions are as follows:

1. How do I continue learning new information while filling in the gaps in my knowledge?
2. How do I recognize when a concept comes in to play, and thus deduce how to apply it?

I will provide an example of how far exactly this weak chain of understanding goes. This last week, I had a last-in-series real analysis midterm, and was asked a question regarding showing that a 'closed rectangle' was compact using the fact that the collection of 'nested rectangles' (a collection such that R_(i+1) is completely contained in R_(i)) had a non-empty intersection. For once, I could draw an example image with each closed rectangle only having two dimensions. However, my lack of understanding with as simple of a concept of compactness caused me to not answer at all. What I mean by 'lack of understanding', is not my misunderstanding of the definition itself, but rather how to apply said definition to reach a goal.

I hope that this is a relatively understandable question for someone out there, and greatly appreciate anyone who takes the time to reply! Thank you :)

Answer 1

A lot of it comes from experience, having seen lots of math, and having done lots of problems. For your particular problem, a thought process leading to you being able to solve the question might be like this.

1. Recall $K$ is compact if given an open cover $\{O_{\alpha}\}$, there is a finite subcover $\{O_{\alpha_j}\}_{j=1}^n$.
2. Look at the question, which talks about closed sets.
3. Think about how closed sets can possibly have anything to do with compactness, which is about open sets.
4. Recall $F$ is closed if and only if $F^c$ is open.
5. Try to rephrase the definition of compactness in terms of closed sets. If $\bigcup_{\alpha} O_{\alpha} \supset K$, then $\bigcap_{\alpha}O_{\alpha}^c \cap K = \varnothing$, and there exists $\alpha_1,\dots,\alpha_n$ such that $\bigcap_{1\leq j \leq n}O_{\alpha_j}^c\cap K = \varnothing$. In other words, we should have $K$ is compact if and only if given any family of closed sets whose intersection does not intersect $K$ implies the intersection of every finite subfamily also does not intersect $K$. Take the contrapositive and we find $K$ is compact if and only if given a family of closed sets, the intersection of every finite subfamily with $K$ is non-empty implies the intersection of the entire family has a non-empty intersection with $K$. Now apply the fact to closed (and bounded) rectangles in $\mathbb R^n$.

The key insight here is really the duality principle between open and closed sets. Given seemingly unrelated hypotheses, try to relate these with other concepts you are familiar with.