The weak Golbach conjecture states that you can break up any odd number into the sum of, at most, three prime numbers. Prime numbers cannot be evenly divided by any other number than themselves or 1.
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Fields Medalist Dr. Terence Tao gives career advice to aspiring mathematicians. It’s sound advice from someone who has excelled in his field for over a decade. Here is Dr. Tao’s wiki page. BTW, he’s 34 years old and became a full professor at UCLA at 23. AMS has got a recent interview with Dr. Tao.
Metric spaces are sets with a measure of distance between each of its elements. Compact spaces are spaces in which each sequence (xn) admits a converging sub-sequence. This is called the Bolzano-Weierstrass property and such compact spaces are called sequencially compact. There is a more general way of defining a compact space, by saying that each of its open covers has a finite subcover.