Cardinality of an intersection

Question

Is it true that given two real intervals $[a, b] $ and $[c, d] $, the cardinality of their intersection is either $0$ (when they're disjoint), $1 $ (when either $ b=c $ or $ d=a$) or $\mathfrak{c}$?

Answer 1

By considering the possible cases ordering between $a$, $b$, $c$, or $d$, you can determine that either the intersection is $\emptyset$, one points, or a closed interval containing a nonempty open interval. $\mathbb{R}$ bijections onto any nonempty open interval. Hence in the latter of the three cases, the cardinality is $2^{\aleph_0} = \mathfrak{c}$.