Number of connected components of a set.

Question

I have to find the number of connected components of the following sets under the usual topology on $\mathbb{R}.$

• $\{ x\in \mathbb{R} : x^5 + 60x \geq 15x^3 + 10x^2 + 20 \}$
• $\{x\in \mathbb{R} : x^3(x^2 + 5x - \frac{65}{3} ) > 70x^2 - 300x - 297\}$

Also I have to find the number of compact components among them.

Basically I am totally new in facing such problems. I know the basic definitions of the topological terms but have no idea in solving such problems.

I will be grateful if someone guides me to the right direction. Thank you.

Answer 1

Both sets are essentially, after rewriting, of the form $\{x \in \mathbb{R}: p(x) \ge 0\}$, with $p(x)$ some polynomial. Or with $< 0$ or $>0$, or $\le 0$.

Determine the zeroes (or at least the number of them), and then the set in question is just a union of intervals (open or closed, depending on strictness on equality) of $\mathbb{R}$. The number of components will be the number of those disjoint intervals. In the $\le 0$ or $\ge 0$ case we could have loose singletons too, as a degenerate closed interval.