I'm looking for a formalisation of the following statement (from Bröcker and tom Dieck's book on the representations of compact groups, page 157). Suppose $T$ and $T'$ are tori in the compact Lie group $G$:
"Since tori are compact and connected, if $T\subsetneq T'$, then dim $T < $ dim $T'$."
I can see intuitively why this statement might be true, but how does one prove it?
If instead they have the same dimension, then $T$ is open in $T'$. Since $T$ is compact, it is closed in $T'$. Thus $T$ is both open and closed in $T'$ and thus $T = T'$ since $T'$ is connected.