Compactness of the set in $\mathbb{R}$

Question

Can anyone explain me is this set compact or not? $$S:=\{x ∈ \mathbb{R} : x ∈ (2, 3] ∪ (4, 5]\text{ or } x=10\}$$ I already know that for instance $(2,3]$ and $(4,5]$ are not compact, but does it imply for the union?

Answer 1

in $\Bbb R$, a set is compact iff it is closed and bounded, $S=(2,3]\cup (4,5]\cup \{10\}$ in this case is not closed, since $2$ as a limit point is not in $S$.

(RECALL: $S$ is closed if the set of all limit point is a subset of $S$)

Answer 2

Asuume that $S$ is compact. Then $(2,3]$ is compact as a closed subset of $S$, a contradiction.

Answer 3

Suppose that $S$ is compact. Then $S$ isclosed, but $S$ does not contain the limit points 2 and 4, a contradiction.

Answer 4

The open cover $\{U_n : n = 1,2,\cdots\}$ has no finite subcover, where $$ U_n = S \cap \left(2+\frac{1}{n},\infty\right) . $$