Define a topological space X that is not compact and define a set A ⊂ X that is compact. Use the definition of finite open subcovers to show that A is compact.
Ok so I think that a topological space that would not be compact could be the set of integers Z on standard topology. A subset could be [-3,3]. But I am not sure how to so that [-3,3] is compact using the definition of open subcovers.
In $\mathbb{Z}$, the interval $[-3, 3]$ - that is, $\{-3, -2, -1, 0, 1, 2, 3\}$ - is compact because it is finite, and any finite subset of any space is compact. Do you see why? HINT: If $A$ is finite, write $A=\{a_1, . . . , a_n\}$; now if $\mathcal{U}=\{U_i: i\in I\}$ is an open cover of $A$, how many elements of $\mathcal{U}$ does it take to cover $\{a_1\}$? $\{a_1, a_2\}$? . . . All of $A$?
Side note: in $\mathbb{R}$, the interval $[-3, 3]$ is also compact, but this is harder to show.