Visualizing $\cap_{i = 1}^\infty A_i = (\cup_{i = 1}^\infty A_i^c)^c$

Question

As the question title suggests, how do I visualize$$\cap_{i = 1}^\infty A_i = (\cup_{i = 1}^\infty A_i^c)^c?$$Let's start with the right-hand side. So I have a bunch of circles representing the $A_i$'s, right? Then I take the complements of each one, so for each $A_i$, the corresponding $A_i^c$ is the entire ambient space with the $A_i$ removed. I have trouble visualizing that taking the union of all those $A_i^c$'s then taking the complement of that union is our desired intersection $\cap_{i = 1}^\infty A_i$. Could anybody help me visualize this?

Answer 1

It’s easier, I think, to let $B_i=A_i^c$, so that your identity becomes $\bigcap_iB_i^c=\left(\bigcup_iB_i\right)^c$. Now draw your picture, with circles for the sets $B_i$. The set $\bigcap_iB_i^c$ consists of those points that are not in any of the circles, and so does the set $\left(\bigcup_iB_i\right)^c$.

Answer 2

It's easier to visualize $4$ than $\infty$, but the principle is the same.

Draw $4$ circles (and make sure the intersection of all $4$ is nonempty). Note that this intersection is the only region that is not outside any of the circles.

Answer 3

A graphical example of Robert Israel's explanation.

Some sets $A_1$, $A_2$, $A_3$, $A_4$ and their intersection $\cap_i^\infty A_i$.

Now we calculate their complementary (only the shaded parts).

Finally we join them and calculate the complementary of the resulting set.

We see that $\cap_i^\infty A_i = \big(\cup_i^\infty A_i^\complement\big)^\complement$