Show that for each $a\in [0, \infty)$, the subspace $[a, \infty)$ is also compact

Question

i) Show that the intervals $(a, \infty)$, $a \in (0, \infty)$ together with $\emptyset$ and $[0, \infty)$ form a topology on $[0, \infty)$.

ii) Show that in this topology $[0, \infty)$ is compact. Show that for each $a \in [0, \infty)$, the subspace $[a, \infty)$ is also compact.

iii) Notice that intersection $\bigcap_{k=1}^\infty [k,\infty)= \emptyset$,why doesn’t this contradict the statement that “the intersection of a decreasing sequence of nonempty compact sets is nonempty.”

this is not homework, it is from a book with no solutions.

i is easy from definition and checking union and intersection.

Answer 1

i) is straightforward.

For ii), note that if $\cal O$ is an open cover of $[a,\infty)$, then at least one element of $\cal O$ must contain the point $a$. If $a=0$, then the only set satisfying this is $[0,\infty)$, and if $a>0$, then there must exists a set of the form $[b,\infty) \in {\cal O}$ where $b < a$ (otherwise $\cup_{U \in {\cal O}} U = (a, \infty)$). Hence there exists a finite cover.

For iii), the statement should be in terms of closed, not compact sets. In this particular topology, the only non-empty compact sets that is closed is $[0,\infty)$.